《矢量:关于空间、时间和数学变换的惊人故事》封面图片,作者:Robyn Arianrhod

向量

Vector

ROBYN ARIANRHOD 拥有广义相对论博士学位,在莫纳什大学教授数学;她目前是莫纳什大学数学学院的一名附属教授。她为感兴趣的普通读者撰写有关数学和科学的文章——她的书已被翻译成多种语言,并入围国家文学奖,她的文章发表在各种媒体上,包括《澳大利亚最佳科学写作》

ROBYN ARIANRHOD has a PhD in the theory of general relativity and taught mathematics at Monash University; she’s currently an Affiliate in Monash’s School of Mathematics. She writes about maths and science for the interested general reader—her books have been translated into several languages and shortlisted for national literary awards, and her articles have appeared in various outlets, including Best Australian Science Writing.

“阿丽亚罗德的向量,是一部科学论述的杰作,读起来就像是一场令人愉悦的认知悬念之旅,它介绍了向量、向量的概括以及与之相伴的数学物理符号工具,所有这些都通过与惊人联系和应用密切相关的历史小插图描述而相互契合。”

‘Arianrhod’s Vector, a masterpiece of science exposition, reads as a welcoming cognitive cliffhanger tour of vectors, their generalizations, and their accompanying symbolic tools of mathematical physics, all dovetailing through germane history vignette accounts of astonishing connections and applications.’

——约瑟夫·马祖尔, 《时钟幻象:我们测量时间的神话》作者

―Joseph Mazur, author of The Clock Mirage: Our Myth of Measured Time

“向量是具有大小和方向的量。这一思想使物理学家和数学家能够以新的维度想象和描述世界。作者追溯了向量在过去 5,000 年中的影响,以及向量(和张量)为何至今仍然具有现实意义。”

‘A vector is quantity that has magnitude and, crucially, direction. This idea has enabled physicists and mathematicians to imagine and describe the world in new dimensions. The author traces the influence of vectors over the past 5,000 years, and why vectors (and tensors) are still relevant today.’

―书商

―The Bookseller

“关于现代物理学的演变有很多书:从牛顿到麦克斯韦到爱因斯坦再到量子理论。但很少有作者关注使这些物理理论成为可能的数学革命。只有当数学工具包从简单的标量扩展到包括四元数、矢量和张量等工具和思想时,物理学家和数学家才能找到语言来描述日益令人困惑的宇宙。阿里安罗德出色地讲述了数学革命背后的故事,这是推动十九世纪和二十世纪物理革命的引擎;这本书非常值得一读。”

‘There have been lots of books about the evolution of modern physics: from Newton to Maxwell to Einstein and on to quantum theory. But seldom does an author pay attention to the mathematical revolutions that made those physical theories possible. Only as the mathematical toolkit expanded from simple scalars to include such tools and ideas as quaternions and vectors and tensors could physicists and mathematicians find the language to describe an increasingly bewildering universe. Arianrhod does a remarkable job telling the story of the mathematical revolution under the hood, the engine that drove the physics revolutions of the nineteenth and twentieth centuries; the result is a book well worth your time.’

—查尔斯·赛菲 (Charles Seife), 《零:一个危险想法的传记》作者

—Charles Seife, author of Zero: The Biography of a Dangerous Idea

在《向量》中,阿丽安罗德以优美的笔触向我们展示了数学并不是只有极客才居住的陌生世界。它我们周围的世界。”

In Vector, Arianrhod shows, with beautiful ease, that maths is not some foreign world only geeks inhabit. It is the world around us.’

―亚当·斯宾塞,《亚当·斯宾塞数字大全》作者:你想知道的关于 1 到 100 的所有数字

―Adam Spencer, author of Adam Spencer's Big Book of Numbers: Everything you wanted to know about the numbers 1 to 100

“如果所有数学都消失了,”物理学家理查德·费曼认为,“这将使物理学倒退整整一周。”数学家马克·卡克反驳道:“倒退整整一周就是上帝创造世界的那一周。”阿丽安罗德说服我们,向量和张量就是上帝创造的产物之一。学生和老师应该一起阅读这本优秀的书。'

‘“If all mathematics disappeared,” physicist Richard Feynman opined, “it would set physics back precisely one week.” To which mathematician Mark Kac retorted, “Precisely the week in which God created the world.” Arianrhod persuades us that vectors and tensors are among those creations. Students and teachers should read this excellent book together.’

—Marjorie Senechal, 《数学情报员》主编

—Marjorie Senechal, editor-in-chief of The Mathematical Intelligencer

“阿丽安罗德善于将复杂的事情简单化,并且善于讲故事,在传达数学之美和力量方面无与伦比。威廉·罗文·汉密尔顿、詹姆斯·克拉克·麦克斯韦和阿尔伯特·爱因斯坦在这个关于一个改变我们世界的简单想法的戏剧性故事中栩栩如生。”

‘With a flair for exposition that makes the complex simple, and a gift for storytelling, Arianrhod is without peer in conveying the beauty, and power, of mathematics. William Rowan Hamilton, James Clerk Maxwell, and Albert Einstein come alive in this dramatic tale of a simple idea that changed our world.’

—阿米尔·亚历山大,《无穷小:一个危险的数学理论如何塑造了现代世界》一书的作者

—Amir Alexander, author of Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World

“每个人都明白以特定速度朝特定方向移动意味着什么。但花了很长时间才开始用一个明确的概念——矢量来思考这种行为。Arianrhod 生动而详细的编年史解释了为什么矢量和张量是我们思考宇宙的最佳方式的核心。”

‘Everyone understands what it means to move at some particular speed in some particular direction. But it took a long time to start thinking of such behavior in terms of a single clarifying concept, the vector. Arianrhod’s lively and detailed chronicle explains why vectors and tensors are at the heart of our best ways to think about the universe.’

—肖恩·卡罗尔,《宇宙中最伟大的想法》一书的作者

—Sean Carroll, author of The Biggest Ideas in the Universe

致摩根

To Morgan

并致以我全部的爱和衷心的感谢

with all my love and heartfelt thanks.

向量

Vector

关于空间、时间和数学变换的惊人故事

A surprising story of space, time, and mathematical transformation

罗宾阿里安罗德

ROBYN ARIANRHOD

新南威尔士大学出版社承认贝德加尔人是新南威尔士大学兰德威克校区和肯辛顿校区所在地未割让领土的传统所有者,并承认他们与国家和文化的持续联系。我们向过去和现在的贝德加尔长者致敬。

UNSW Press acknowledges the Bedegal people, the Traditional Owners of the unceded territory on which the Randwick and Kensington campuses of UNSW are situated, and recognises their continuing connection to Country and culture. We pay our respects to Bedegal Elders past and present.

新南威尔士大学出版社出版的书

A UNSW Press book

发布者

Published by

新南方出版公司

NewSouth Publishing

新南威尔士大学出版社有限公司

University of New South Wales Press Ltd

新南威尔士大学

University of New South Wales

悉尼新南威尔士州 2052

Sydney NSW 2052

澳大利亚

AUSTRALIA

https://unsw.press/

https://unsw.press/

© 2024 Robyn Arianrhod。保留所有权利。

© 2024 by Robyn Arianrhod. All rights reserved.

首次由芝加哥大学出版社在美国出版

First published in the United States by The University of Chicago Press

本书受版权保护。除《版权法》允许的出于私人学习、研究、批评或评论目的的合理使用外,未经书面许可,不得以任何方式复制本书的任何部分。如有疑问,请联系出版商。

This book is copyright. Apart from any fair dealing for the purpose of private study, research, criticism or review, as permitted under the Copyright Act, no part of this book may be reproduced by any process without written permission. Inquiries should be addressed to the publisher.

澳大利亚国家图书馆提供了该书的目录记录

国际标准书号 9781761170089 (平装本)
  9781742239002 (电子书)
  9781742239965(电子PDF)

设计Lauren Michelle Smith

Design Lauren Michelle Smith

封面图片Jim.belk,Wikimedia Commons

Cover image Jim.belk, Wikimedia Commons

本书已尽一切合理努力获得版权资料的使用许可,但在某些情况下无法追踪版权。作者欢迎提供这方面的信息。

All reasonable efforts were taken to obtain permission to use copyright material reproduced in this book, but in some cases copyright could not be traced. The author welcomes information in this regard.

图像

内容

CONTENTS

序幕

Prologue

(1)代数的解放

(1)The Liberation of Algebra

(2)微积分的到来

(2)The Arrival of Calculus

(3)向量的创意

(3)Ideas for Vectors

(4)了解空间(和存储)

(4)Understanding Space (and Storage)

(5)令人意外的新玩家和非常缓慢的接受度

(5)A Surprising New Player and a Very Slow Reception

(6)泰特和麦克斯韦:孵化电磁矢量场

(6)Tait and Maxwell: Hatching the Electromagnetic Vector Field

(7)从四元数到向量的缓慢旅程

(7)The Slow Journey from Quaternions to Vectors

(8)向量分析的最终成果——以及四元数的“战争”

(8)Vector Analysis at Last—and a “War” over Quaternions

(9)从空间到时空:向量的新转折

(9)From Space to Space-Time: A New Twist for Vectors

(10)弯曲空间和不变距离:通往张量的途径

(10)Curving Spaces and Invariant Distances: On the Way to Tensors

(11)张量的发明及其重要性

(11)Inventing Tensors—and Why They Matter

(12)一切汇聚在一起:张量和广义相对论

(12)Everything Comes Together: Tensors and the General Theory of Relativity

(13)接下来发生了什么

(13)What Happened Next

结语

Epilogue

时间线

Timeline

致谢

Acknowledgments

笔记

Notes

指数

Index

序幕

PROLOGUE

时不时地,我们对世界的理解就会出现惊人的突破。例如,颠覆了我们对太阳系中心位置的信念,相对论革命改变了我们看待时间和空间的方式,也改变了我们看待自己在宇宙中的位置的方式。电磁无线电波的惊人发现带来了无线技术,并对我们的日常生活产生了奇妙的改变,还有量子革命,它带来了看似神奇的新微技术,并以超自然的方式打破了我们对“现实”的观念。当然,还有数字革命,它仍在改变我们彼此交流的方式,尤其是在人工智能变得如此复杂的今天。这些突破带来了新的技术和文化时代,关于它们的文章也很多。然而,鲜为人知的是,这些科学和技术范式转变与同样引人注目的数学革命齐头并进。这是一个关于一些无名革命的故事:它们背后令人着迷的想法,以及使它们成为可能的人。

Every now and then, there’s a spectacular breakthrough in our understanding of the world. The upending of the belief in our central place in the solar system, for example, and the relativistic revolution that changed the way we see time and space—and the way we see our place in the cosmos itself. The dramatic discovery of electromagnetic radio waves, which led to wireless technology and its wondrous transformation of our everyday lives—and the quantum revolution, with its seemingly magical new microtechnologies and its preternatural shattering of our notions of “reality.” And then, of course, there’s the digital revolution that is still changing the way we communicate with each other, especially now that AI has become so sophisticated. These breakthroughs have brought into being new technological and cultural eras, and much has been written about them. It’s less well known, however, that these scientific and technological paradigm shifts went hand in hand with equally dramatic mathematical revolutions. This is a story about some of those unsung revolutions: the fascinating ideas behind them, and the people that made them possible.

从本质上讲,我在这里要讲述的故事是关于人类记录和理解我们周围所有数据的方式的演变。特别是,我将探索戏剧性的数学变换,这些变换为我们提供了称为“向量”和“张量”的非凡概念,因为它们是很多现代科学,以及我们的很多技术,都离不开这些语言。这些语言帮助我们揭开了宇宙的奥秘,就像我们是神一样。

At its core, the story I’ll tell here is about the evolution of the way we humans record and make sense of all the data that swirl around us. In particular, I’ll explore the dramatic mathematical transformations that gave us the remarkable concepts called “vectors” and “tensors,” for they underlie so much of modern science—and much of our technology, too. They are languages that have helped us uncover mysteries of the universe as if we were gods.

向量和张量具有如此强大威力的原因之一是,我们能够以一种全新且透明的方式处理空间维度,而这反过来又使我们能够发现新的自然法则以及这些法则的新技术应用。任何时候,只要你想利用空间中的位置,就需要处理这些维度,比如旋转机械臂、设计桥梁或风力涡轮机;弄清楚电动机或发电机中电磁力的影响,或者预测电磁波、水波甚至引力波的路径;绘制卫星轨迹或校准 GPS 等导航系统;或者你需要在空间或时空中进行分析的几乎任何事情。

One reason for this awesome power is that vectors and tensors made it possible to handle the dimensions of space in a new and transparent way— and this, in turn, made it possible to discover new laws of nature, and new technological applications of these laws. Anytime you want to utilise locations in space you need to handle these dimensions—rotating a robot arm, say, or designing a bridge or wind turbine; figuring out the effect of an electromagnetic force in a motor or generator, say, or predicting the path of an electromagnetic wave, a water wave, or even a gravitational wave; plotting the trajectory of a satellite or calibrating a guidance system such as GPS; or just about anything you need to analyse in space or space-time.

随着故事的展开,我们将更详细地了解向量和张量的物理威力。但这些语言不仅仅涉及物理维度,它们还涉及信息的“维度”。你可能读过关于“大数据”和信息革命的文章,但正是向量和张量帮助使数据变得可用和可理解——就像化学元素周期表既是化学中的组织工具又是理论工具一样,只不过我们故事中的数学应用范围要广泛得多。

We’ll see in more detail the physical power of vectors and tensors as the story unfolds. But these languages are not just about physical dimensions— they’re about “dimensions” of information, too. You’ve likely read about “big data” and the information revolution, but it is vectors and tensors that help make data usable and comprehensible—the way the periodic table of chemical elements is both an organisational and theoretical tool in chemistry, except that the math in our story is so much more widely applicable.

然而,向量和张量本身非常简单——至少从表面上看——因为你确实可以简单地将它们视为表示信息的简洁方式。例如,你可能还记得在学校学过,向量可以编码有关物理量(例如速度或力)的大小方向的信息。因此,你可以用指向所需方向的箭头来表示它,而箭头的长度给出大小或“幅度”。张量增加了更多信息层,因此它们更像是多维数组而不是箭头。但是,当数学家发现这些箭头和数组如何相互组合的规则时,他们意识到他们已经找到了一种思考全新思想的全新语言。这是一个相当奇妙的想法。

Yet vectors and tensors themselves are remarkably simple—on the face of it, at least—for you can, indeed, begin by thinking of them simply as neat ways of representing information. For instance, you might remember from school that a vector can encode information about both the size and the direction of a physical quantity—say, a velocity or a force. So, you can represent it with an arrow pointing in the required direction, while the arrow’s length gives the size or “magnitude.” Tensors add in more layers of information, so they are like multidimensional arrays rather than arrows. But when mathematicians discovered the rules for how these arrows and arrays combine with each other, they realised they had found a brand-new language for thinking brand-new thoughts. And this is a rather wonderful idea.

一个简单的例子就是,几千年来,数学家只与数字打交道。实数的演化数字系统已经足够引人注目了,但这些数字只表达一件事:数量——重量、高度、距离、金额、苹果数量等等。另一方面,向量和张量可以同时编码多个事物,这就是为什么它们是如此出色的表示大量数据的方式。这些额外的信息意味着,与单个数字相比,向量和张量可以提供更丰富的工业或 IT 问题图景,或者物理模型。

A simple illustration of what I mean by this is that for thousands of years mathematicians worked only with numbers. The evolution of real number systems was remarkable enough, but these numbers express only one thing: quantity—the magnitude of a weight, height, distance, amount of money, number of apples, and so on. Vectors and tensors, on the other hand, encode several things at once, which is why they are such great ways of representing a lot of data. And this extra information means that vectors and tensors can offer a far richer picture of an industrial or IT problem, say, or a physical model, than a single number ever could.

第一个认识到向量语言威力的著名物理学家是十九世纪略显古怪的苏格兰领主詹姆斯·克拉克·麦克斯韦。我们稍后会正式介绍他,但他的电磁理论是第一个现代场论,他利用该理论破解了光的本质这一长期谜题,并预测了无线电波的存在——这一切都是一举成功。他最初的理论直观上是“向量的”,但当他了解到向量实际上是“一个东西”,有自己的数学规则时,他意识到它们是更简洁、更优雅地表达他的发现的正确工具。

The first major physicist to recognise the power of vector language was the gently eccentric nineteenth-century Scottish laird James Clerk Maxwell. We’ll meet him properly later, but his theory of electromagnetism was the very first modern field theory, which he used to crack the longstanding riddle of the nature of light and to predict the existence of radio waves—all in one fell swoop. His initial theory was intuitively “vectorial,” but once he learned that vectors were actually “a thing,” with their own mathematical rules, he realised they were the right tools for expressing his discovery more succinctly and elegantly.

起初,并没有多少人认真对待他:正如我们将看到的,他突破性地应用“矢量场”来表示自然界的电磁场……对于主流物理学家来说,这实在是太数学化、太“非物理”了。但是,如果麦克斯韦凭借其对矢量的出色应用获得认可已经如此困难,那么想象一下这种数学语言的创造者一定拥有的热情和自信。

Not that many people took him seriously at first: as we’ll see, his breakthrough application of “vector fields” to represent nature’s electromagnetic fields was … well, just too mathematical, too “unphysical,” for mainstream physicists. But if it was difficult enough for Maxwell to achieve recognition for his superb application of vectors, imagine the passion and self-belief the creators of this mathematical language must have had.

这个故事中的众多明星之一是爱尔兰数学家威廉·罗文·汉密尔顿。他是“向量”一词的创造者,也是第一个提出向量数学理论的人——他马上就意识到,他创造了一个如此新颖的东西,打破了数学家们几千年来理所当然的规则。然而,当他瞥见这种新语言的可能应用时,在麦克斯韦发表他的奇妙理论的六年前,他欣喜若狂。他欣喜地写信给一位与他一起研究这个问题的同事,“还有什么比这更简单、更令人满意的吗?你不觉得,也不认为,我们走在正确的道路上,以后会得到感谢吗?别介意什么时候……” 1汉密尔顿不仅在说

One of the many stars of this story is the Irish mathematician William Rowan Hamilton. He’s the one who coined the term “vector” and first presented its mathematical theory—and he knew right away that he’d created something so new it broke a rule mathematicians had taken for granted for thousands of years. Yet when he glimpsed the possible applications of this new language, six years before Maxwell published his marvelous theory, he was over the moon. He wrote joyously to a colleague who was working with him on the subject, “Could anything be simpler or more satisfactory? Do you not feel, as well as think, that we are on a right track, and shall be thanked hereafter? Never mind when....”1 Hamilton was speaking not only

这里是关于向量的,还有他发明的四元数,四维的包含向量的“数字”。正如我们所见,四元数可以完成向量所能做的一切,但它们在航天器制导和图像处理等特定任务的编程中效率更高,这只是两个现代应用。

about vectors here, but also his invention of quaternions, four-dimensional “numbers” that contain vectors. As we’ll see, quaternions can do everything that vectors can do, but they are more efficient in programming certain tasks in spacecraft guidance and image processing, to mention just two modern applications.

可怜的汉密尔顿从未得到足够的感谢:他于 1865 年去世,就在麦克斯韦发表电磁学理论几个月后,但他还没能将其改写成完整的矢量语言。2

Poor Hamilton never did receive sufficient thanks: he died in 1865, just a few months after Maxwell published his theory of electromagnetism but before he’d been able to recast it into full vector language.2

至于张量,麦克斯韦在张量被发明之前就去世了,但我敢打赌他肯定也意识到了张量的威力。麦克斯韦去世的那一年正是爱因斯坦出生的那一年,这尤其具有象征意义——不仅因为爱因斯坦的理论受到了麦克斯韦理论的启发,还因为爱因斯坦对张量所做的贡献与麦克斯韦对矢量所做的贡献一样:他是第一位展示张量实际威力的物理学家。张量使他能够创造弯曲的时空和现代宇宙学学科——也使他能够预测引力波和透镜的存在,并准确量化引力对时间的影响,而现在正是利用这种影响,让 GPS 导航变得如此精确。

As for tensors, Maxwell died before they were developed, but I’m betting he would have recognised their power, too. He died the same year that Einstein was born, which is especially symbolic—not only because Einstein’s theories were inspired by Maxwell’s, but also because Einstein did for tensors what Maxwell had done for vectors: he was the first major physicist to show their practical power. They enabled him to create curved space-times, and the discipline of modern cosmology—and they enabled him to predict the existence of gravitational waves and lenses, and to accurately quantify the gravitational effect on time that is now used to make GPS directions so accurate.

实验物理学家花了四分之一世纪的时间在实验室中验证了麦克斯韦对无线电波的预测,而探测爱因斯坦的引力波则花了一百年的时间。这表明这些基于矢量和张量的理论是多么领先于时代。这种预测能力是数学语言令人兴奋的地方之一。就好像用数学描述物理现实的行为创造了一个放大镜,通过数学模式揭示了长期隐藏的潜在物理属性。我们将在后面看到具体的例子,但在这里我只想补充一点,量子理论也很好地利用了矢量和张量——到目前为止,它的任何预测都没有被推翻。

It took experimental physicists a quarter of a century to verify in the lab Maxwell’s prediction of radio waves, and it took a hundred years to detect Einstein’s gravitational waves. That’s an indication of how far ahead of the game these vector- and tensor-based theories were. This kind of ability to make predictions is one of the exciting things about mathematical language. It’s as if the act of describing physical reality mathematically creates a magnifying glass, revealing, through mathematical patterns, underlying physical attributes that had long lain hidden. We’ll see specific examples of this as we go, but here I’ll just add that quantum theory also makes fine use of vectors and tensors—and, so far, none of its predictions have been disproved.

由于向量和张量是存储和使用信息的方式,它们的用途当然远不止在物理学中。正如我之前所暗示的,它们在越来越多需要处理大量数据的领域中发挥着重要作用——从工程和遗传学到搜索引擎和人工智能,以及介于两者之间的更多领域。

Since vectors and tensors are ways of storing and using information, they’re useful far more widely than in physics alone, of course. As I intimated earlier, they are playing a fundamental role in a growing number of areas that need to handle a lot of data—from engineering and genetics to search engines and artificial intelligence, with much more in between.

然而,这些数学思想的全部威力的发展是如此惊人,影响如此深远,以至于我将它们的发现视为革命。将张量视为向量的概括是有帮助的,但这只是后见之明:从一种刚刚起步的向量语言发展到一种以严格方式包含向量和张量的复杂语言,花了三百年时间。而要获得向量概念的第一个新生迹象,已经花费了许多世纪——事实上,如果我们追溯到现存最古老的数学记录,则花费了数千年时间。因为向量和张量的历史与数据符号表示的历史息息相关,这些古代文献表明,找到表示信息的方法是数学故事的核心。

Yet the development of the full power of these mathematical ideas was so astonishing, and so far-reaching, that I’m treating their discoveries as revolutions. It’s helpful to think of tensors as a generalisation of vectors, but that is hindsight: it took three hundred years to move from a fledgling form of vector language to a sophisticated language that incorporates vectors and tensors in a rigorous way. And to get to that first, nascent hint of the vector concept, it had already taken many centuries—millennia, in fact, if we go back to the oldest surviving mathematical records. For the history of vectors and tensors is linked with the history of the symbolic representation of data, and these ancient documents show that finding ways to represent information is at the heart of the story of mathematics itself.

因此,我将从简要回顾这一切的起源开始讲述这个故事。当然,我在这里以及在后面章节中对这段漫长历史的讲述不可能详尽无遗。它必然是有选择性和主观性的。我的目标之一只是展示复杂的数学思想需要多长时间以及需要多少跨文化合作才能发展起来。这是一条漫长而曲折的道路,通往如今广泛使用的现代向量和张量分析,我想讲述的故事是一段思想之旅——这些思想往往令人惊讶,有时平淡无奇,但从一开始,它们就一直充满着可能性。

So, I’ll begin this tale with a brief tour back to the beginning of it all. Of course, my telling of this long history, here and as the story unfolds in later chapters, cannot be exhaustive. It’s necessarily selective and subjective. One of my goals is simply to show just how long it takes—and how much intercultural cooperation is needed—for sophisticated mathematical ideas to develop. It was a long and winding road to the modern vector and tensor analyses that are so widely used today, and the story I want to tell is a journey of ideas—ideas that are often surprising, sometimes mundane, but which have always been, right from the beginning, rich with possibility.

不过,如果在阅读本书的任何时候,你发现自己想浏览一下细节并继续阅读故事,那也没问题!

Still, if at any point throughout the book you find you want to skim the details and get on with the story, that’s fine, too!

回到起点

BACK TO THE BEGINNING

大约五千年前,生活在现今伊拉克附近的人们开始通过在粘土盘或粘土片上刻出楔形符号来记录信息。这些奇怪的符号被称为“楔形文字”,能够记录和控制商品和土地等有形物品交换的经济和行政力量一定令人惊叹。但又过了一千年——在手指、鹅卵石以及算盘和桌子等计算工具的帮助下——抽象的数字系统和算术规则才得以发展。

It was some time about five thousand years ago that people living in the area around present-day Iraq began to write down information by scratching wedge-shaped signs into clay discs or sheets. These strange signs are known as “cuneiform” script, and the economic and administrative power of being able to record and control the exchange of tangible things such as goods and land must have seemed amazing. But it took another thousand years—and the help of computational tools such as fingers, pebbles, and eventually abaci and tables—for abstract number systems and the rules of arithmetic to develop.

楔形文字数学的发明者生活在底格里斯河和幼发拉底河之间的肥沃平原上——一千年后,希腊人将这片地区称为美索不达米亚(意为“两河之间”)。这里孕育了许多相互关联的文化,因此“美索不达米亚”一词至今仍被广泛用于描述在这个农业发达、知识丰富的地区发展起来的古代数学和其他文化创新。当然,这里并不是唯一一个从简单计数转向复杂数值算术的地方。但我们对美索不达米亚数学的了解远远超过对其他早期文化的了解,这仅仅是因为许多非凡的粘土文献幸存了下来。

The inventors of this cuneiform mathematics lived on the fertile plains between the Tigris and Euphrates Rivers—an area that the Greeks, a thousand years later, would call Mesopotamia (or “between two rivers”). It hosted a number of linked cultures, so the term “Mesopotamian” is still generally used to describe the ancient mathematical and other cultural innovations developed in this agriculturally and intellectually rich region. Of course, it was not the only place to move from simple counting to sophisticated numerical arithmetic. But we know so much more about Mesopotamian mathematics than that of other early cultures, simply because so many of those remarkable clay documents survived.

一些最早的、更为复杂的数字和数学泥板,可以追溯到近四千年前,上面写着乘法表。这些东西存在了这么久似乎令人惊讶,但当然,你需要能够进行加法和乘法才能完成基本的经济任务。历史学家已经洞察了这些早期任务的性质,因为这些功能也记录在泥板上。值得注意的是,其中一些最古老的文献包含表格,列出了各种正方形或矩形区域的边长,每个边长都与其面积相匹配——这种表格布局后来会演变成数学矩阵。简单的信息列表会演变成向量——但稍后我们会更多地讨论所有这些数学变化,因为它将向我们展示数学家如何从简单的会计列表转变为建模诸如电磁波或量子计算机所依赖的量子比特等复杂事物。与此同时,这些古老的表格对于计算潜在的粮食产量、种子需求、耕种田地所需的劳动力数量以及需要支付的工资和税款至关重要——这些都是任何大型社会为生产和分配食物和其他必需品所需要的东西。3

Some of the earliest of the more sophisticated numerical and mathematical tablets, dating from nearly four thousand years ago, contain multiplication tables. It might seem surprising that such things have been around for so long, but of course you need to be able to add and multiply to carry out basic economic tasks. Historians have gained insight into the nature of those early tasks, because these functions, too, were recorded on clay tablets. Tellingly, some of the oldest of these documents contain tables giving lists of the lengths of the sides of various square or rectangular fields, each matched with their areas—the kind of tabular layout that would later morph into mathematical matrices. Simple lists of information would morph into vectors—but more on all this mathematical morphing later, for it will show us how mathematicians went from simple accounting lists to modeling such complex things as electromagnetic waves, for example, or the qubits that underpin quantum computers. Meantime, these ancient tables were vital for working out potential grain harvests, seed requirements, the amount of labour needed to work the fields, and the wages and taxes to be paid—the kinds of things that any large society needs in order to create and distribute food and other necessities.3

在最早的美索不达米亚先进社会中,经济正常运转所需的土地面积估计并不需要精确。测量员可以用木桩和绳子划出田地,然后测量其边长,但他们不必担心土地是否完全呈矩形,因为无论如何,土地都归国家所有。然而,大约公元前 1900 年,情况开始发生变化,普通民众也很快拥有了土地——这意味着测量需要变得更加准确,因为土地纠纷即将开始其漫长的历史。(相比之下,许多原住民一直沿用旧方法,直到殖民者打破了平衡。)因此,在最早的乘法表帮助会计师计算近似正方形和矩形面积的数百年后,美索不达米亚的测量员已经研究出如何制作完美的 90° 角——这表明他们可能在毕达哥拉斯之前一千年就发现了“毕达哥拉斯定理”。

In the earliest of these advanced Mesopotamian societies, the estimates of field sizes needed for the economy to run properly didn’t need to be exact. Surveyors could mark out fields with pegs and ropes, and then measure their sides, but they didn’t have to worry about making the land parcels exactly rectangular because the state owned it all anyway. From about 1900 BCE, however, things began to change, and soon ordinary folk could own land, too—and this meant that surveying needed to become more accurate, because land disputes were soon to begin their long history. (By contrast, many First Nations people practiced the old way right up until colonisation disrupted the balance.) And so, hundreds of years after the earliest multiplication tables had helped accountants calculate the areas of approximate squares and rectangles, Mesopotamian surveyors worked out how to make perfect 90° corners—which suggests they may have discovered “Pythagoras’s theorem” a millennium before Pythagoras.

一代又一代的小学生都在念诵这条古老的规则:“斜边的平方等于两邻边的平方和”。毕达哥拉斯生活在公元前六世纪,但大约 3700 年前的楔形文字泥板(最著名的是标有 Plimpton 322 和 Si.427 的泥板)包含了令人信服的证据,证明这条规则比它的希腊同名规则古老得多。Plimpton 322 泥板(图 0.1)已经破损,但残存的碎片列出了十五对与直角三角形的对角线和短边有关的数字。所使用的数字的选择以及表格各列的标题表明,这可能是一个隐含的、六十进制的“毕达哥拉斯三元组”列表,正如它们现在的名称一样,它是一组测量员可以从中选择的整数三元组。例如,(3, 4, 5) 是毕达哥拉斯三元组,因为 3 2 + 4 2 = 5 2。石板 Si.427 支持这种解释,因为它是将一块土地私人细分为矩形和三角形的平面图——每个都有完美的矩形角和尺寸,符合毕达哥拉斯定理。4

Generations of schoolchildren have chanted this ancient rule: “the square of the hypotenuse equals the sum of the squares of the two adjacent sides.” Pythagoras lived in the sixth century BCE, but cuneiform tablets dating from around 3,700 years ago—most famously those labeled Plimpton 322 and Si.427—contain compelling evidence that the rule is much older than its Greek namesake. The Plimpton 322 tablet (fig. 0.1) has been broken, but the surviving fragment lists fifteen pairs of numbers relating to the diagonal and the shorter side of a right-angled triangle. The choice of numbers used, and the headings in the table’s columns, suggest that this was probably a list of implicit, sexagesimal “Pythagorean triples,” as they are now called—a set of integer (whole number) triples that surveyors could choose from. For example, (3, 4, 5) is a Pythagorean triple, because 32 + 42 = 52. The tablet Si.427 supports this interpretation, for it is a plan of the private subdivision of a land parcel into rectangular and triangular fields—each with perfectly rectangular corners and dimensions that fit Pythagoras’s theorem.4

一千年后,古希腊数学家开始对测量一系列角度感兴趣,而不仅仅是 90°,因为他们不仅想测量地球,还想测量星星。天文学和数学一样,都是最古老的科学。毕竟,广阔而耀眼的夜空是一件奇妙的事情。由于他们无法测量到星星的距离,这些古希腊天文学家发现,你可以通过测量它们的角度来精确定位它们——因此他们还发现了两个关键的东西。首先,三角学。这并不是说早期的文明没有某种形式的角度制表和“三角”计算——历史学家仍在争论和解释楔形文字。但现存最古老的明确三角表存在于克劳狄斯·托勒密 1850 年前的希腊数学天文学汇编《天文学大成》中。(如果你忘记了三角学的基本思想,你可以提前看第 3 章中的图 3.4。)其次,希腊人发展了用坐标表示空间位置的思想——这一杰出的创新与新兴的矢量思想有很大关系。

A thousand years later, ancient Greek-speaking mathematicians were interested in measuring a range of angles, not just 90°, because they wanted to survey not only the earth but the stars, too. Along with mathematics, astronomy is the oldest science. After all, a vast and dazzling night sky is a wondrous thing. Since they had no way of measuring the distance to the stars, these ancient Greek astronomers figured out that you could pinpoint them by measuring their angles—and so they also discovered two key things. First, trigonometry. Which is not to say that earlier cultures didn’t have some form of angular tabulation and “trigonometric” calculation, too— historians are still debating and interpreting cuneiform texts. But the oldest extant, explicit trig table survives in Claudius Ptolemy’s extraordinary 1,850-year-old compilation of Greek mathematical astronomy, Almagest. (If you’ve forgotten the basic idea of trigonometry, you can look ahead to fig. 3.4 in chap. 3.) Second, the Greeks developed the idea of representing positions in space with coordinates—a brilliant innovation that has a lot to do with the emerging idea of a vector.

图像

图 0.1 . 普林普顿 322。这块引人注目的石碑被标记为 322,位于哥伦比亚大学的 GA 普林普顿收藏中。不幸的是,这块石碑也代表着殖民时期的掠夺,因为普林普顿于 1922 年从一位考古学家兼文物商人手中买下了它。摄影师不详。Wikimedia Commons,公共领域。

FIGURE 0.1. Plimpton 322. This remarkable tablet is labeled 322 in the G. A. Plimpton Collection at Columbia University. Unfortunately, this tablet also represents colonial looting, because Plimpton bought it from an archaeologist-cum-antiquities dealer in 1922. Photographer unknown. Wikimedia Commons, public domain.

不过,在我进一步讨论这个问题之前,我想先在接下来几段中介绍一下与我们的故事相关的其他古代数学文化。例如,讲希腊语的托勒密生活在罗马统治下的埃及亚历山大——这提醒我们,随着帝国的扩张和衰落,残酷的动乱往往会随之而来,也提醒我们,在数学史上,“希腊”指的是讲希腊语的数学家的作品,无论他们的种族、民族或居住地如何。这也提醒我们,著名的亚历山大女数学家希帕蒂娅在三个世纪后对《天文学大成》进行了评论。虽然我强调现代数学的多元文化历史,但我应该指出,《天文学大成》是一个阿拉伯语单词,意思是“最伟大的”,这个词长期以来一直是用来描述托勒密编纂的非正式形容词,以区别于当时的其他作品。这本真正“最伟大”的书之所以能流传至今,很大程度上要归功于中世纪的阿拉伯语译者和注释者,因此,它自那时起就以阿拉伯语名称为人所知,这是再合适不过的了。

Before I talk a little more about this, though, in the next couple of paragraphs I want to acknowledge the other ancient mathematical cultures that also connect with our story. For instance, Greek-speaking Ptolemy lived in Alexandria, Egypt, under Roman rule—a reminder of the brutal upheavals that tend to follow as empires expand and fall, and also of the fact that in math history “Greek” refers to the work of Greek-speaking mathematicians, no matter their race, ethnicity, or where they lived. A reminder, too, of the famous female Alexandrian mathematician Hypatia, who wrote a commentary on Almagest three centuries later. And while I’m emphasising modern mathematics’ multicultural history, I should note that Almagest is an Arabic word meaning “the greatest,” a term that had long been the informal adjective used to describe Ptolemy’s compilation, to distinguish it from others of the time. This truly “greatest” book survives today largely because of its medieval Arabic translators and annotators, so it is fitting that it has been known ever since by its Arabic name.

和美索不达米亚人一样,古埃及人早在五千年前就开始用象形文字来表示数字,他们也对测量土地很感兴趣。当然,希腊人并不是第一个测量天空的人:美索不达米亚人、埃及人和许多其他古代民族都曾绘制过月亮和星星的轨迹,以便他们能够设计日历、祭拜神灵、穿越陆地和海洋,并注意到重要的巧合——比如,每年黎明前,当明亮的天狼星升起时,尼罗河就会开始泛滥。

Like the Mesopotamians, ancient Egyptians had begun to represent numbers some five thousand years ago—in their case, in hieroglyphic symbols—and they, too, were interested in surveying the land. And of course the Greeks were not the first to survey the sky: the Mesopotamians, Egyptians, and many other ancient peoples had charted the paths of the moon and stars so that they could devise calendars, propitiate gods, navigate their way across land and sea, and notice important coincidences—such as the fact that each year the Nile would begin to flood when the bright star Sirius rose just before dawn.

埃及和美索不达米亚的天文记录似乎是算术记录而非三角学记录——它们记录的是天体事件之间的天数,而不是不同时间恒星和行星的角度位置。但这些记录非常准确,以至于后来讲希腊语的天文学家很乐意使用它们。当时的大多数文化都没有留下天文观测的书面记录,但有些仍然以故事的形式流传下来——例如托雷斯海峡岛民传说中的 Baidam(或 Beizam),天鲨。这种“鲨鱼”是希腊人称为“熊”的星座,又名大熊座或北斗七星,它在夜空中的位置告诉人们何时播种庄稼——以及何时远离大海,因为鲨鱼正在繁殖!然后还有巨石阵、纽格兰奇和其他古代天文学仪式建筑,包括澳大利亚瓦萨鲁恩县的 Wurdi Youang 石圈,它可能比中东和欧洲的同类石圈早数千年。中国人、印度人、玛雅人、印加人和许多其他人也发展出了令人印象深刻的天文学知识——2017 年,国际天文学联合会通过在新的恒星命名系统中纳入八十六个土著星名,承认了所有这些古代文化的贡献。5

It seems that Egyptian and Mesopotamian astronomical records were arithmetical rather than trigonometrical—they charted the number of days between celestial events, rather than the angular positions of stars and planets at different times. But these records were so accurate that later Greek-speaking astronomers were happy to make use of them. Most cultures from this time did not leave written records of their astronomical observations, but some still survive in stories—such as the Torres Strait Islander tradition of Baidam (or Beizam), the celestial shark. This “shark” is the constellation the Greeks called “the bear,” aka Ursa Major or the Big Dipper, and its position in the night sky told the people when to plant crops—and when to stay out of the sea because the sharks were breeding! Then there’s Stonehenge, New Grange, and other ancient astronomically aligned ceremonial structures, including the Wurdi Youang stone circle in Wathaurong country, Australia, which is likely thousands of years older than its Middle Eastern and European counterparts. The Chinese, Indians, Mayans, Incas, and many other people, too, developed an impressive knowledge of astronomy—and in 2017, the International Astronomical Union acknowledged the contributions of all these ancient cultures by including eighty-six indigenous star names in a new star naming system.5

至于古代对数学的贡献,我们将在接下来的两章中看到更多。但现在让我回到坐标,以及它们与向量的关系。这让我想起了托勒密,他给我们留下了最早的复杂记录,不仅是数学天文学,还有使用坐标来表示空间位置的记录。我不能把所有的功劳都归给他,无论是他的天文学还是他的坐标——辉煌的想法需要很长时间,需要很多人的贡献才能实现。但托勒密非常善于给予他的前辈应有的认可——这对他们来说也很好,因为他们的大部分工作后来都因霉菌、害虫、事故、战争而丢失了,而且由于《天文学大成》非常成功,早期的作品被丢弃了。6亚历山大的欧几里得是一个显著的例外,因为他著名的《几何原本》的内容完整地保存了下来。欧几里得在公元前 300 年左右撰写了这本书,比托勒密的《天文学大成》早了大约 450 年,它为我们提供了至今仍在使用的理解日常(平坦)“欧几里得”空间的规则。

As for ancient contributions to mathematics, we’ll see a little more of those in the next two chapters. But now let me return to coordinates, and how they relate to vectors. Which brings me back to Ptolemy, who left us the earliest sophisticated record not just of mathematical astronomy, but also of the use of coordinates to represent positions in space. I cannot give him all the credit, for his astronomy or for his coordinates—it takes a long time, and contributions from many people, for brilliant ideas to come to fruition. But Ptolemy was very good at giving his forerunners due recognition—which is just as well for them, since most of their work was later lost to mold, pests, accidents, war, and the fact that Almagest was so successful that the earlier works were discarded.6 Euclid of Alexandria is a notable exception, for the content of his famous Elements survives in full. Euclid composed this book around 300 BCE, some 450 years before Ptolemy’s Almagest, and it gave us the rules we still use for understanding everyday (flat) “Euclidean” space.

然而,就现存的记录而言,托勒密引入了纬度和经度的概念,我们今天用它来定位地球上的位置。在他的著作《地理学》中,他列出了数千个地理位置——在《天文学大成》中,他有一个包含一千多颗恒星的星表,这些恒星是通过纬度和经度的天体类似物定位的。所有这些坐标都是角度——为了测量它们,托勒密描述了如何用阴影棒和像量角器一样的刻度盘来找到太阳的角度,而为了定位其他恒星,他使用了浑天仪。(浑天仪是一个复杂的可移动环网络,形成一个骨架球体,包围着一个微小的地球;这些环代表了太阳的视年和视日轨道、观察者的地平线和子午线——它们与观察者的纬度和经度有关——以及一个标有度数刻度的环和另一个标有观测星空的孔。)

As far as the surviving records go, though, it is Ptolemy who introduced the idea of latitude and longitude that we use today for locating places on Earth. In his book Geography, he lists thousands of geographical locations—and in Almagest, he has a star catalogue of more than a thousand stars, located via celestial analogues of latitude and longitude. All these coordinates are angles—and to measure them, Ptolemy described how to find the sun’s angle with a shadow stick and a graduated dial like a protractor, while to locate the other stars he used an armillary sphere. (This is a complex network of movable rings forming a skeletal globe that encloses a tiny Earth; the rings represent such things as the apparent yearly and daily orbits of the sun, the observer’s horizon and meridian—which are related to the observer’s latitude and longitude—as well as a ring marked with a degree scale and another with sighting holes to the stars.)

又过了一千二百年,我们才逐渐在数学课堂上习惯使用方格纸上的坐标,特别是通过十四世纪法国数学家的工作。Nicole Oresme(尽管与欧几里得同时代的人阿波罗尼乌斯也曾略微领悟过这一思想)。三个世纪后,另一位法国人勒内·笛卡尔更接近这一思想,后来的数学家们效仿他,牢固地建立了人们熟悉的x - y “笛卡尔”坐标系,其轴线在中间相交,即网格的“原点”。现在,在这个简短的初步概述中,我们将着重介绍矢量的概念。

It took another twelve hundred years to get an inkling of the kind of coordinates on graph paper that we’re used to in math classes today— particularly through the work of the fourteenth-century French mathematician Nicole Oresme (although Euclid’s near-contemporary Apollonius of Perga had also glimpsed the idea). Three centuries later, another Frenchman, René Descartes, came closer still, and, following his lead, later mathematicians firmly established the familiar x-y “Cartesian” coordinates, whose axes meet in the middle, at the “origin” of the grid. And now we are homing in, in this short preliminary overview, on the idea of a vector.

因此,我想用本介绍的剩余部分来进一步介绍现代向量是什么,以及它们能做什么。不过,首先,我想重申我在前面几页中概述的向量悠久、多元文化史前史的重要性,因为本书的大部分内容将集中在发明微积分、向量和张量的现代欧洲人的工作上。今天,数学是一项真正普遍的事业,但在我们故事的大部分发生的十九世纪,英国(当时包括爱尔兰)特别适合成为科学强国。它拥有历史悠久的大学,以及越来越多的机械学院和工人学院,用于成人教育。 (1874 年,职业女性终于有了自己的学院,稍后我们会介绍一些在第一所女子大学学习的女性。)英国还拥有完善的银行系统,可以进行创业技术投资;拥有不断发展的铁路网络,可以促进交流和市场发展;拥有丰富的原材料——英国自己的煤炭、水和钢铁,以及从殖民地掠夺的材料。这些因素支撑了十八和十九世纪的工业革命,科学和技术齐头并进,相互促进——当然,不仅仅是在英国。欧洲也有同样发达的机构和科学界——尤其是我们的故事中的法国、意大利和德国。然而,正如我们所看到的,以及我们将在接下来的两章中看到的,这些现代科学界很大程度上归功于他们古代和中世纪的多元文化先驱。

So, I want to spend the rest of this introduction showing a little more about what modern vectors are, and what they can do. First, though, I want to reiterate the importance of vectors’ long, multicultural prehistory that I sketched in the preceding pages, because most of this book will focus on the work of the modern European men who invented calculus, vectors, and tensors. Math today is a truly universal endeavor, but in the nineteenth century when much of our story takes place, the United Kingdom (which then included Ireland) was especially well placed to be a scientific powerhouse. It had long-established universities along with a growing number of Mechanics’ Institutes and Working Men’s Colleges for adult education. (Working women finally got their own college in 1874, and later we’ll meet some of the women who studied at the first University Women’s Colleges.) It also had a well-established banking system that enabled entrepreneurial technological investment, it had a growing railway network that facilitated communication as well as markets, and it had abundant raw materials—its own coal, water, and steel, as well as materials plundered from its colonies. These factors underpinned the Industrial Revolution of the eighteenth and nineteenth centuries, in which science and technology went hand in hand, each fueling the other—and not only in Britain, of course. Europe had similarly well-developed institutions and scientific communities—especially, for our story, France, Italy, and Germany. As we’ve seen, though, and as we’ll see in the next two chapters, these modern scientific communities owed much to their ancient and medieval multicultural forebears.

快速展望:想象新维度、新世界

A QUICK LOOK AHEAD: IMAGINING NEW DIMENSIONS, AND NEW WORLDS

你可能还记得,在学校里,你可以用画箭头来表示从原点O到坐标网格上的点P的方向,例如参见下图 0.2。这是一个向量的示例或模型——它是一个“位置”向量,用P的方向和与O的距离来表示 P 的位置。但是,编码不止一个东西(不止一个数字)的想法实际上相当复杂,这个故事将带我们了解这一切是如何产生的。我们将看到,创建具有量级和方向的单个符号的想法花了很长时间。

You may remember from school that you can represent the direction from the origin O to a point P on a coordinate grid by drawing an arrow, as in figure 0.2 below. This is an example or model of a vector—it’s a “position” vector, representing P’s position in terms of its direction and distance from O. But the idea of something that encodes more than one thing— something beyond a single number—is actually quite sophisticated, and this story will take us through how it all came to be. And as we’ll see, it took a very long time to create the idea of a single symbol having both magnitude and direction.

因此,接下来的几页只是简要的回顾,也许会提醒您两个基本思想,您可能已经了解过,但没有意识到它们是如此具有开创性。我们稍后会更详细地介绍它们,但我在这里提到它们是为了为我们在旅途中需要的一些术语和概念打下基础——同时也让您了解为什么向量在今天如此重要。张量是向量的多维表亲,将在故事的后面出现,因为它们将建立在为向量开发的概念和符号体系之上——而向量之旅出乎意料地是分层的。但随着我们探索这些层次,我们也将建立支撑张量的思想。

So, the next couple of pages are just a quick look ahead, and perhaps a reminder of two basic ideas you may have learned but didn’t realise were so groundbreaking. We’ll meet them in more detail later, but I mention them here to lay tracks for some terminology and concepts we’ll need on the journey—and also to give a glimpse into why vectors are so important today. Tensors, vectors’ multidimensional cousins, will come much later in the story, for they will build on the concepts and symbolism developed for vectors—and the journey to vectors is unexpectedly layered. But as we explore these layers, we’ll also be building up ideas that underpin tensors, too.

第一个想法是符号。数学符号有时看起来很神秘,所以我在这本书中的目标之一就是与你分享一些概念上的突破,这些突破使得为新想法发明新符号成为必要。即使是那些熟悉的xy也是经过了很长时间的酝酿,我们将在下一章中看到。我们不会在接下来的几章中开始介绍现代矢量符号的诞生——这表明这些想法的演变需要多长时间。不过现在,我只会将这个故事与你可能已经知道的内容联系起来。例如,位置向量通常表示为rr, 或者r^即使你不熟悉矢量,你也可能记得看到过教科书将这样的符号与普通字母和数字混淆。这是因为粗体类型(或字母顶部的“帽子”或箭头或字母下方的波浪线)是当今表示矢量的常用方式,以将它们与表示数字、大小而没有方向的普通字母区分开来。因此,在这种情况下,没有粗体类型或帽子或其他标记的r仅表示距离的大小,而不表示方向;像这样的量被称为“标量”,而不是“矢量”。另一个例子该符号的其中一种表示方式是, v代表速度,其中包括方向和速率,v代表速率——速度的大小。

The first idea is notation. Mathematical symbols can sometimes seem mysterious, so one of my aims in this book is to share with you some of the conceptual breakthroughs that made it necessary to invent new symbols for new ideas. Even those familiar x’s and y’s were a long time coming, as we’ll see in the next chapter. We won’t get to the inauguration of modern vector symbolism for several chapters—which goes to show just how long it took for these ideas to evolve. For now, though, I’ll just lay tracks that connect this story with what you may already know. For example, position vectors are often denoted by r, r, or r^. Even if you’re not au fait with vectors, you might remember seeing textbooks mixing up signs like this with ordinary letters and numbers. That’s because boldface type—or “hats” or arrows on top of letters or squiggles underneath them—are common ways of denoting vectors today, to distinguish them from ordinary letters that represent numbers, magnitudes without direction. So, in this case r, with no bold type or hat or other marker, would represent just the magnitude of the distance, with no reference to the direction; quantities such as this are called “scalars,” as opposed to “vectors.” Another example of this notation is v for velocity, which includes direction as well as speed, and v for speed—the magnitude of the velocity.

图像

图 0.2。P位置向量r,其分量为r xr y。r 的量级箭头的长度,表示从OP的距离)用 | r | 或r表示,它可利用毕达哥拉斯定理从分量中得出。

FIGURE 0.2. P’s position vector r, with components of magnitude rx and ry. The magnitude of r—the length of the arrow, representing the distance from O to P—is denoted by |r| or r, and it is found from the components using Pythagoras’s theorem.

第二个关键概念是矢量有“分量”。在书的一页或一张方格纸的二维空间中,有两个独立的空间方向(通常表示为xy方向或轴),因此点有两个坐标,它们之间的箭头有两个分量。换句话说,分量是从每个轴测量箭头时得到的值,如图 0.2所示。

The second key notion is that a vector has “components.” In the twodimensional space of a page in a book, say, or a piece of graph paper, you have two independent spatial directions—commonly represented as the x and y directions or axes—so your points have two coordinates and the arrow between them has two components. In other words, components are the values you get for your arrow when you measure it from each axis, as you can see in figure 0.2.

同样,在三维空间中,有三个独立的方向和三个分量。然后是时空,其中有四个坐标轴,三个空间轴和一个时间轴,而你的隐喻箭头有四个分量——所以你的位置矢量将测量不仅要测量行进的距离,还要测量所用的时间。这可能比简单地测量三个空间分量并从手表上读取时间分量要复杂一些——因为如果你相对于要测量的对象移动,时间和空间坐标会以混合的方式发生变化。当你处于引力场中时,情况会更加复杂——这是广义相对论而非狭义相对论的领域。我稍后会更多地谈论相对论,但在这里我只想指出时空中的矢量有四个分量。 (我还应该指出,空间和时间的这种混合是为什么在现代物理学论文中“spacetime”的拼写可能比“space-time”更常见的原因,因为它表明空间和时间是相互交织的——它们不仅仅是两个不同的东西互相嫁接在一起。但这两种拼写在今天的数学和物理学教科书中都有使用,而连字符版是该概念的创始人赫尔曼·闵可夫斯基和爱因斯坦使用的。我们稍后会介绍闵可夫斯基和爱因斯坦。)

Similarly, in three-dimensional space, there are three separate directions and three components. Then there’s space-time, where you have four coordinate axes, three spatial ones and a time axis, and your metaphorical arrows have four components—so your position vector would measure not only the distance traveled but also the time it took. It can be a bit trickier than simply measuring the three spatial components and reading off the time component from your wristwatch—because time and space coordinates change, in an intermingled way, if you’re moving relative to what you want to measure. It’s even more complicated when you’re in a gravitational field—the province of general rather than special relativity. I’ll talk more about relativity later, but here I just want to note that vectors in spacetime have four components. (I should also note that this intermingling of space and time is why the spelling “spacetime” is perhaps more common than “space-time” in modern physics papers, because it shows that space and time are interwoven—they are not just two different things grafted on to each other. But both spellings are used in maths and physics texts today, and the hyphenated version is the one used by the founder of the concept, Hermann Minkowski, and by Einstein. We’ll meet Minkowski and Einstein later.)

这个故事中极具争议的一件事就是箭头和其组成部分之间的细微差别。当人们最终掌握了这个微妙之处时,不仅为现代本科矢量数学的创造打开了大门,而且也为张量的创造打开了大门。我们将看到,产生混淆的部分原因是您也可以通过列出它们的组成部分来表示矢量,这比画箭头更容易、更经济——尤其是当您想脱离三维空间时。例如,假设您以 35 英里/小时的速度向北行驶(沿着一条平坦的二维街道——数学家们喜欢从简化假设开始探索新想法!),并且您选择南北方向作为y轴。从几何上讲,您的速度矢量将在图 0.2中沿y轴向上指向,并且它的长度为 35 个单位。但您也可以将此速度写为v = (0, 35),这意味着您沿y轴以 35 英里/小时的速度移动,沿x轴以 0 英里/小时的速度移动。因此,标量 35 表示您的速度(单位为英里/小时),矢量 (0, 35) 表示速度朝北 ( y ) 方向。(用矢量符号标记 (0, 35),如v = (0, 35),可将其与笛卡尔网格上的单个点区分开来 - 因此您可以看到符号的重要性。)您可以像这样表示任何速度,无论方向如何。例如,如果你以每小时 35 英里的速度向东北行驶,你的矢量分量将是三十五2三十五2(正如我在本尾注7中所展示的)。

One of the things that will prove extremely controversial in this story is the subtle difference between the arrow, so to speak, and its components. When the subtlety is finally grasped, the way will open for the creation not just of modern undergraduate vector maths, but of tensors, too. We’ll see that the confusion arose partly because you can also represent vectors simply by listing their components, and this is easier and more economical than drawing an arrow—especially when you want to move out of three dimensions. For example, suppose you’re driving at 35 mph in a northerly direction (along a flat, two-dimensional street—mathematicians love to explore new ideas by starting with simplifying assumptions!), and you choose the north-south direction to be the y axis. Geometrically, your velocity vector would point upward along the y axis in figure 0.2, and it would be 35 units long. But you can also write this velocity as v = (0, 35), meaning you’re moving 35 mph along the y axis and 0 mph along the x axis. So, the scalar, 35, gives your speed (in units of mph), and the vector, (0, 35), shows that the speed is in the northerly (y) direction. (Labeling (0, 35) with a vector symbol, as in v = (0, 35), distinguishes it from a single point on a Cartesian grid—so you can see the importance of notation.) You can represent any velocity like this, no matter the direction. For example, if you were traveling, say, northeast at 35 mph, your vector components would be 352,352 (as I’ve shown in this endnote7).

图像

图 0.3 . 半径为r的圆上点 P 的极坐标;θ 以逆时针方向测量。速度分量在r方向和 θ 增加的方向上测量,如图所示。

FIGURE 0.3. Polar coordinates of a point P on a circle of radius r; θ is measured anticlockwise. The velocity components are measured in the r direction and in the direction of increasing θ, as shown.

更一般地,暂时跳过从数字到符号代数的漫长道路,你可以用(v x,v y,v z)表示三维空间中的速度分量,这表示你在x方向以v x英里每小时的速度行进,在y方向以v y英里每小时的速度行进,在 z 方向以v z英里每小时的速度行进。更一般地,三维空间中的速度分量可以用(v 1v 2v 3)表示,因为笛卡尔x,y,z坐标系不是唯一可用的坐标系。例如,在二维中,极坐标(r,θ)可用于定位圆周上的点,如图0.3所示;矢量(v 1v 2)表示你在r方向以v 1英里每小时的速度行进,在 θ 方向以v 2英里每小时的速度行进。要定位球面上的某个点,可以使用球面坐标 (r, θ, φ) 将其扩展到三维,其中 φ 是矢量与z轴之间的角度。但细节在这里并不那么重要:这种符号背后的故事才是如此有趣——当然还有应用。

More generally—and skipping blithely, for the moment, over the long road from numbers to symbolic algebra—you might represent velocity components in three-dimensional space by (vx, vy, vz), indicating that you’re going vx mph in the x direction, vy mph in the y direction, and vz mph in the z direction. Even more generally, the components of velocity in 3-D space can be denoted by (v1, v2, v3), because the Cartesian x, y, z coordinate system is not the only one available. For instance, in two dimensions polar coordinates (r, θ) are useful for locating points on the circumference of a circle, as you can see in figure 0.3; the vector (v1, v2) would mean you’re traveling v1 mph in the r direction, v2 mph in the θ direction. To locate a point on a sphere, you can extend this to three dimensions using spherical coordinates (r, θ, φ), where φ is the angle between the vector and the z axis. But the details are not so important here: it’s the story behind this symbolism that is so interesting—and, of course, the applications.

例如,在四维时空中,爱因斯坦将他的坐标标记为(x 1x 2x 3x 4),其中前三个是通常的空间坐标,第四个是时间坐标。因此,该系统中四维速度矢量(或“四速度” 8)的分量将被标记为(v 1v 2v 3v 4)。

For example, in four-dimensional space-time, Einstein labeled his coordinates as (x1, x2, x3, x4), where the first three are the usual spatial coordinates and the fourth is the time coordinate. So, the components of a four-dimensional velocity vector (or “four-velocity”8) in this system would be labeled (v1, v2, v3, v4).

以此类推,我们可以想象任意数量的维度。将向量中的信息表示为一串分量的美妙之处就在于此:如果你可以写出(v 1v 2v 3v 4),那么有什么能阻止你写出(v 1v 2v 3v 4v 5,...)任意数量的(有限!)维度呢?这正是弦理论家在提出我们的宇宙可能不只有四个维度,而是十个、十一个甚至二十六个维度时所做的事——有几种不同的弦理论。你可以将额外的维度视为使空间能够容纳微小弦振动的多种方式,每种方式代表不同的基本粒子或力。

By analogy, it’s possible to imagine any number of dimensions. This is the beauty of representing the information in a vector as a string of components: if you can write (v1, v2, v3, v4), what’s to stop you writing (v1, v2, v3, v4, v5, …) for as many (finite!) dimensions as you like? That’s just the kind of thing string theorists have done when they suggest our universe might have not four but ten or eleven or even twenty-six dimensions—there are several different string theories. You can think of the extra dimensions as enabling space to accommodate the many ways in which tiny strings can vibrate, each way representing a different fundamental particle or force.

弦理论是数学“向量空间”中的维度可以表示比普通空间和时间更奇特事物的一个例子。在量子力学中,电子的磁取向轴或“自旋”可以是“向上”或“向下”——或者是两者的“叠加”,就好像电子无法决定它希望自旋向上还是向下。它类似于二维速度矢量,其中一个分量在x方向上,另一个分量在y方向上,因此对于电子自旋,你可以让两个自旋方向“向上”和“向下”作为轴。

String theory is an example of the way the dimensions in a mathematical “vector space” can represent more exotic things than ordinary space and time. In quantum mechanics, to take another example, the axis of the magnetic orientation, or “spin,” of an electron can be “up” or “down”—or a “superposition” of the two, as if the electron can’t decide whether it wants its spin to be up or down. It’s analogous to a two-dimensional velocity vector having one component in the direction x, say, and another in the direction y, so for electron spin you can let the two spin directions “up” and “down” be your axes.

同样,在计算中,你用两个二进制数字(“位”)0 和 1 来编码信息,它们在物理上可以用例如打开和关闭电开关来表示。在量子计算中,位的类似物是量子比特(发音为“cue-bit”,是“量子比特”的缩写)。量子 0 和 1 可以在物理上表现为电子自旋的两个基本状态——0 可以用自旋“向上”表示,1 可以用自旋“向下”表示——因此,量子比特也可以在数学上表示为二维矢量。

Similarly, in computing you encode information with two binary digits (“bits”), 0 and 1, which are represented physically by, for example, turning an electric switch off and on. In quantum computing, the analogue of a bit is the qubit (pronounced “cue-bit,” short for “quantum bit”). Quantum 0’s and 1’s can be manifested physically as the two fundamental states of electron spin—0 might be represented by spin “up” and 1 by spin “down”—so qubits, too, can be represented mathematically as two-dimensional vectors.

向量在现代商业和技术应用中也很有用。例如,轴或“维度”可能代表网站问卷或政治调查中的不同问题,或影响房价或其他社会经济数据,或图像处理中像素的位置和颜色。顺便说一句,如果你熟悉图像处理,你就会知道在某些计算应用程序中,“矢量”一词的用法与我一直以来概述的通常数学含义略有不同。图像处理中的“矢量文件”使用方程式来生成形状,而不是使用矢量来存储有关像素等的信息。这些“矢量文件”中生成的形状确实沿着形状从一点“携带”到另一点的线,“载体”是“矢量”的本义。将数学矢量看作箭头,你可以看到它们也将线从箭头的尾部带到箭头的尖端。尽管如此,箭头,或者更一般地说是其组成部分的字符串,是大多数应用程序(物理、商业计算)中使用的矢量类型。

Vectors are useful in modern business and tech applications, too. For instance, the axes or “dimensions” might represent different questions on a website questionnaire or political survey, or the different factors affecting house prices or other socioeconomic data, or the positions and colours of pixels in image processing. By the way, if you’re familiar with image processing, you’ll know that in some computing applications, the word “vector” can be used slightly differently from the usual mathematical meaning that I’ve been outlining. The “vector files” in image processing use equations to generate shapes, rather than vectors for storing information about such things as pixels. The shapes produced in these “vector files” do “carry” lines from point to point along the shape, and “carrier” is the original meaning of “vector.” Looking at mathematical vectors as arrows, you can see how they, too, carry the line from the tail to the tip of the arrow. Still, the arrow, or more generally the string of its components, is the kind of vector used in most applications—in physics, business, and computing.

但是威廉·罗文·汉密尔顿 (William Rowan Hamilton) 并没有因为可以写下任意数量的组成部分来表示各种物理、数字和经济系统或场景而感到兴奋。自从人类能够计数和书写以来,存储和表示数据一直很重要,但计算也是如此 - 向量和张量使您能够同时进行表示和计算。这就是它的魔力所在:尽管将向量写为组成部分列表简单且实用,但汉密尔顿如此兴奋的是,当你将向量视为一个整体,而不仅仅是其各个组成部分时,向量算术规则使向量比单纯的数字更强大。

But William Rowan Hamilton didn’t get so excited just because you can write down any number of components to represent various physical, digital, and economic systems or scenarios. Storing and representing data has been important ever since humans could count and write, but so has computation—and vectors and tensors enable you to do both the representing and the calculating at the same time. That’s where the magic comes into it: despite the simplicity and utility of writing vectors as lists of components, what Hamilton was so thrilled about was that when you consider the vector as a whole, not just its individual components, the rules of vector arithmetic make vectors far more powerful than numbers alone.

例如,将麦克斯韦方程写成全矢量形式,可以相对容易地推导出电磁波方程——麦克斯韦曾预言过电磁波的存在。磁场矢量和矢量积在数学上定义电子等亚原子粒子的自旋方面也发挥着作用(再举一个例子),从而使自旋能够应用于现实世界——例如,在磁共振成像(MRI)中,它用于通过“观察”患者体内来诊断患者。

For example, writing Maxwell’s equations in whole vector form makes it relatively easy to deduce the equations for electromagnetic waves—a phenomenon whose very existence Maxwell had predicted. Magnetic field vectors and vector products also play a role in mathematically defining the spin of subatomic particles such as electrons (to take just one more example), thereby enabling spin to be applied in the real world—in magnetic resonance imaging (MRI), say, which is used to diagnose patients by “seeing” inside their bodies.

汉密尔顿做梦也想不到这样的事情。但是,一旦他的矢量算法问世,它的模式就引起了其他数学家的兴趣——尤其是那些喜欢看看他们能把逻辑推向多远的人。数学规则和语法的后果。有时他们会走得太远,以至于陷入一种新的现实,就像爱丽丝穿过镜子掉进兔子洞一样。

Hamilton couldn’t have dreamed of such things. But once his vector arithmetic was out in the world, its patterns intrigued other mathematicians—especially those who like to see how far they can push the logical consequences of the rules and grammar of mathematics. And sometimes they push so far that they tumble into a new kind of reality, like Alice through the looking glass and down the rabbit hole.

打破规则

BREAKING THE RULES

并不是说爱丽丝的创造者刘易斯·卡罗尔(Lewis Carroll,又名牛津大学数学家查尔斯·道奇森)是这次创造性推动和颠覆的人之一。甚至有人猜测,爱丽丝梦游仙境中的疯帽子茶会之所以如此荒诞可笑,是因为它是对汉密尔顿打破规则的向量的戏仿——尽管从表面上看,很难判断道奇森是喜欢它们还是认为它们可笑。9无论如何,当汉密尔顿发现与普通算术不同,在向量算术中存在多种执行乘法的方法时,他几乎让所有人都望而却步。此外,正如我们将看到的,向量乘法并不总是遵循长期确立的数值乘法规则。

Not that Alice’s creator, Lewis Carroll—aka Oxford mathematician Charles Dodgson—was one of those doing the creative pushing and tumbling in this case. It has even been conjectured that the Mad Hatter’s Tea Party in Alice’s Adventures in Wonderland is so delightfully absurd because it is a parody of Hamilton’s rule-breaking vectors—although on the face of it, it’s hard to tell whether Dodgson was delighted by them or thought them ridiculous.9 Either way, Hamilton pulled the rug out from almost everyone when he discovered that, unlike ordinary arithmetic, there is more than one way to carry out multiplications in vector arithmetic. What’s more, vector multiplication doesn’t always follow the long-established rules of numerical multiplication, as we’ll see.

汉密尔顿关于乘法的看似荒谬的发现,以及随后所有突破性的矢量和张量发展和应用,都需要使用代数符号。我说的是“矢量算术”,但更一般地说是“矢量代数”。简单地说,算术与数字有关,而代数​​与字母有关,字母代表数字或可量化的量,如温度或速度。这种符号使数学家能够概括。例如,用符号表示普通算术的基本规则(如交换律,a + b = b + aa × b = b × a)可以让你看到适用于所有数字的模式,这样你就不必像最早的数学家那样单独列出每个例子。汉密尔顿尽可能遵循这些基本算术模式,从而发展了他的矢量代数规则,因此当他意识到必须拓宽乘法规则时,他和其他人一样感到震惊。

Hamilton’s seemingly absurd discovery about multiplication, and all the breakthrough vector and tensor developments and applications that followed, required the use of algebraic symbolism. I spoke of “vector arithmetic,” but more generally this is “vector algebra.” Speaking loosely, arithmetic works with numbers, and algebra works with letters that symbolise numbers or quantifiable quantities, such as temperature or speed. This symbolism enables mathematicians to generalise. For example, writing the basic rules of ordinary arithmetic symbolically—such as the commutative law, a + b = b + a and a × b = b × a—enables you to see patterns that hold for all numbers, so you don’t have to list each example separately the way the earliest mathematicians had to do. Hamilton had developed his rules of vector algebra by following, as best he could, these basic arithmetic patterns, so he was as shocked as anyone when he realised he had to broaden the rules for multiplication.

但符号代数思维的出现已经很久了——远远晚于埃及人和美索不达米亚人、古希腊人和中国人、中世纪阿拉伯人和印度人,以及所有其他数学17 世纪之前的文化。即使在 19 世纪,许多数学家也不信任纯粹抽象的符号代数,因为它似乎与日常经验脱节。因此,为了介绍汉密尔顿、麦克斯韦和其他矢量先驱将面临的挑战,下一章将从这一概述回到一个故事,讲述人们最初如何开始用字母表示和计算——以及这个看似简单的步骤如何使数学家能够更有创意地思考。

But symbolic algebraic thinking had been a long time coming—long after the Egyptians and Mesopotamians, the ancient Greeks and Chinese, the medieval Arabs and Indians, and all the other mathematical cultures before the seventeenth century. Even in the nineteenth century many mathematicians mistrusted purely abstract symbolic algebra, so unmoored from everyday experience did it seem. So, to set the scene for the challenges that Hamilton, Maxwell, and the other vector pioneers would face, the next chapter moves back from this overview to a story about how people began to represent and calculate with letters in the first place—and how this seemingly simple step has enabled mathematicians to think more creatively.

不过,首先,我们将站在汉密尔顿的角度,重温他疯狂而超越常理的那一刻:古老的数学规则终究是可以被打破的。

First, though, we’ll walk in Hamilton’s shoes for a mile or two, reliving the moment he had his wild, transgressive realisation that the ancient rules of mathematics could, after all, be broken.

(1)代数的解放

(1) THE LIBERATION OF ALGEBRA

每年 10 月 16 日,数学爱好者都会从邓辛克天文台出发,穿过田野,步行到皇家运河。运河有许多交叉口,但他们特别要前往其中一条:布鲁姆桥。他们正在庆祝汉密尔顿日——爱尔兰最伟大的数学家威廉·罗恩·汉密尔顿——他们正在重现数学史上最著名的一次散步。他们正在纪念 1843 年 10 月 16 日,当天,爱尔兰皇家天文学家汉密尔顿正前往主持爱尔兰皇家科学院的会议。

Every year on October 16, mathematically inclined Dubliners take a communal walk across the fields from Dunsink Observatory and down to the Royal Canal. The canal has many crossings, but they’re heading for one in particular: Broome Bridge. They are celebrating Hamilton Day—that’s William Rowan Hamilton, Ireland’s greatest mathematician—and they’re reenacting one of the most famous walks in the history of mathematics. They are commemorating October 16, 1843, when Hamilton—Ireland’s royal astronomer—was on his way to preside over a meeting of the Royal Irish Academy.

汉密尔顿的妻子海伦也加入了他的散步,但尽管环境很美,尽管他被浪漫主义所吸引,但这并不是传统意义上的浪漫散步:他太专注了。多年来,他一直在努力解决一个看似棘手的问题,这个问题一直困扰着他:如何在三维空间中用代数表示几何运算,比如旋转——操纵机器人、逼真的计算机图像或航天器所需的旋转,汉密尔顿并没有考虑这些高科技的可能性。他感兴趣的只是解决数学问题3-D 旋转,所以他仍在尝试发明最终有助于实现这些应用的数学。然后,当他经过布鲁姆桥时,他突然想到。你只能用 4-D 数学来做 3-D 旋转。1

Hamilton’s wife, Helen, had joined him on his walk, but lovely as the setting was and much as he was drawn to Romanticism, this was not a romantic walk in any traditional sense of the word: he was much too preoccupied. He’d been wrestling for years with a seemingly intractable problem that simply would not leave him alone: how to represent algebraically geometrical operations such as rotations, in three-dimensional space—the kind of rotations you need to manipulate a robot, a realistic computer image, or a spacecraft, say, not that Hamilton was thinking about such high-tech possibilities. He was interested simply in solving the mathematical problem of 3-D rotations, so he was still trying to invent the maths that would eventually help make these applications possible. And then, as he passed by Broome Bridge, it hit him. You could only do 3-D rotations with 4-D maths.1

这个见解已经足够吸引人了,但还不止于此。为了使他的四维数学成立,汉密尔顿必须改变乘法的基本规则。你不能再假设交换律a × b = b × a,这个律在普通数相乘时显然是正确的。你可能还记得与向量乘法的“右手定则”搏斗,但我们将在第 4 章中正确地了解它,因为它提供了一种简洁的方式来证明向量积不交换。然而,学校和大学的向量分析是事后才知道的。回到 1843 年,这种打破所谓交换律的行为是惊人的大胆,甚至是冒昧的:在什么样的宇宙中说 2 × 3 的某个类似物不等于 3 × 2 是有意义的?也许在刘易斯·卡罗尔的《仙境》中,疯帽子告诉爱丽丝,“说出你的意思”并不等同于“你想说的话”,因为“你也可以说‘我看到我吃的东西’和‘我吃我看到的东西’是一回事!” 2但这在数学家几千年来一直居住的直觉精神世界中肯定说不通。汉密尔顿非常兴奋,他拿出小刀,当场在石桥上刻下了他的魔法公式。

This was an intriguing enough insight, but there was more. To make his 4-D maths work, Hamilton had to change the fundamental rules of multiplication. No longer could you assume the commutative law a × b = b × a, which is so obviously true when you multiply ordinary numbers. You might remember wrestling with the “right-hand rule” for vector multiplication, but we’ll meet it properly in chapter 4, for it gives a neat way of showing that vector products aren’t commutative. School and college vector analysis is hindsight, however. Back in 1843, this breaking of the so-called commutative law was astoundingly daring, even presumptuous: In what universe did it make sense to say that some analogue of 2 × 3 does not equal 3 × 2? Perhaps in Lewis Carroll’s Wonderland, where the Mad Hatter tells Alice that “say what you mean” is not the same as “mean what you say,” for “you might just as well say that ‘I see what I eat’ is the same thing as ‘I eat what I see’!”2 But it certainly did not make sense in the intuitive mental world mathematicians had inhabited for several thousand years. Hamilton was so excited that he took out his penknife and scratched his magic formula right then and there on the stone bridge.

这幅传奇数学涂鸦的创作地,是现代人在汉密尔顿日的目的地。原作早已被风雨侵蚀,但我们从汉密尔顿的信件中知道它曾经存在过——在过去的半个世纪里,一块牌匾标记着他灵感迸发的地点,正如他后来回忆的那样,灵感就像电路闭合,火花迸发。2019 年,桥边人行道上的一幅纪念艺术作品以这个电气隐喻为基础,用电光照亮了这幅著名涂鸦的实质,

The site of this legendary piece of mathematical graffiti is the destination of those modern walkers on Hamilton Day. The original carving has long been lost to the wind and rain, but we know it existed from Hamilton’s letters—and for the past half century, a plaque has marked the spot where he experienced the inspiration that came to him, as he later recalled, as if an electric circuit closed and a spark flashed forth. A 2019 commemorative artwork in the pavement by the bridge built on this electrical metaphor, illuminating with electric light the substance of the famous graffiti,

i2 = j2 = k2 = ijk = −1

i2 = j2 = k2 = ijk = −1.

如果汉密尔顿能看到的话!3

If only Hamilton could have seen it!3

你可能已经注意到,汉密尔顿公式中的i、jk是“虚数”,因为没有“实数”的平方可以等于负数。(当你计算一个实数的平方时,你总是将两个同号的数相乘,所以你得到的平方总是正数。这意味着如果你得到的平方是负数,如i 2 = −1,那么这个数i就不是实数,而是所谓的虚数。)

You might have already spotted that the i, j, and k in Hamilton’s formulae are “imaginary” numbers, for no “real” number squared can equal a negative number. (When you’re taking the square of a real number, you’re always multiplying two numbers of the same sign, so you always get a positive square. Which means that if you get a negative square, as in i2 = −1, then such a number i is not a real number; rather, it’s a so-called imaginary one.)

稍后我将探讨汉密尔顿方程的含义,但目前,我想强调的是世界各地的朝圣者继续参观布鲁姆桥的原因。这组看似简单的方程包含了一种新型四维结构的关键,它开辟了一种全新的数学语言,如今已成为各种领域不可或缺的一部分。汉密尔顿将他的四维作品命名为“四元数”;它们包含两部分,一个一维实数,他称之为“标量”,以及一个具有大小和方向的三维(三分量)量,他称之为“矢量”。如果你熟悉现代矢量,那么出于实际目的,你可以用同样的方式思考汉密尔顿的矢量。(我们稍后会看到它们之间微妙的概念差异。它将在我们的故事中扮演一个有争议的角色。)

I’ll explore the meaning of Hamilton’s equations later, but for the moment, I want to emphasise the reason pilgrims from around the world continue to visit Broome Bridge. This seemingly simple line of equations contained the key to a new kind of four-dimensional construct, which opened up a whole new mathematical language that has become indispensable in all sorts of fields today. Hamilton gave the name “quaternions” to his 4-D creations; they contained two parts, a 1-D real number, which he called a “scalar,” and a 3-D (three-component) quantity having magnitude and direction, which he called a “vector.” If you’re familiar with modern vectors, then for practical purposes you can think of Hamilton’s vector in just the same way. (We’ll see the subtle conceptual difference between them later. It will play a controversial role in our story.)

在那次传奇般的行走之后的二十年,詹姆斯·克拉克·麦克斯韦 (James Clerk Maxwell) 创立了电磁场的矢量场理论。而更为复杂的四元数则等了将近 150 年才找到我之前开始列出的现实世界应用——机器人技术、CGI、分子动力学、手机屏幕的旋转、航天器控制等等。例如,月球漫步者尼尔·阿姆斯特朗 (Neil Armstrong) 就非常了解四元数在飞机和航天器导航中发挥的作用。他既是航空工程师,也是宇航员,在生命的最后阶段,他在访问都柏林期间向汉密尔顿表达了自己的敬意。4

Two decades after that storied walk, James Clerk Maxwell created his vector field theory of electromagnetism. The more complicated quaternions had to wait nearly 150 years to find the real-world applications I began to list earlier—robotics, CGI, molecular dynamics, the rotations of our mobile phone screens, spacecraft control, and much more. Moonwalker Neil Armstrong, for one, knew all about the role quaternions now play in airplane and spacecraft navigation. He was an aeronautical engineer as well as an astronaut, and toward the end of his life he paid his own personal tribute to Hamilton during a visit to Dublin.4

在我们进一步了解汉密尔顿是如何成功完成他的探索的——以及四元数和向量能做什么之前——我想告诉你,他是如何设想出虚数和高维数学的。在序言中,我跳过了从三维到四维或更多维的转变,但这种思维转变远非一帆风顺。这是因为几千年来,数学基本上只有两种,算术和几何。算术是关于计算具体的数量,例如金钱、重量和距离——几何也是通过点、线、平面和形状以直观的方式呈现的,你可以在二维页面上绘制它们,也可以在三维空间中表示它们。如果我们想用物理模型模拟我们日常物理现实经验中可以轻易想象到的事物,那么三维就是我们能达到的极限。

Before we take a closer look at how Hamilton succeeded in his quest— and at what you can do with quaternions and vectors—I want to tell you how it came about that he could envisage such things as imaginary numbers and higher-dimensional maths. In the prologue, I skipped through the move from three dimensions to four or more, but this transition in thinking was far from straightforward. That’s because for thousands of years there had essentially been only two kinds of mathematics, arithmetic and geometry. Arithmetic was about counting concrete quantities, such as money, weight, and distance—and geometry, too, was visualised in a palpable way, through points, lines, planes, and shapes that you could draw on a two-dimensional page or represent in three-dimensional space. Three dimensions are as far as we can go if we want to physically model the kinds of things we can readily imagine with our everyday experience of physical reality.

那么,汉密尔顿的四维数学又如何呢?你如何进行这样的数学运算——它如何表示像在普通空间中旋转的物体这样具体的东西?这是一个很长的故事,因为数学家花了很长时间才学会抽象思维——将他们的想象力从有形事物中解放出来。要做到这一点,他们首先必须学会符号思维——从具体的算术和几何转向抽象的代数符号世界。所以,如果你曾经为代数而苦恼,请振作起来:这段悠久的历史表明,一代又一代的著名数学家也曾这样做过。

So, what of Hamilton’s 4-D maths? How can you do such maths—and how can it represent something as concrete as an object rotating in ordinary space? It’s quite a long tale, because it took a very long time for mathematicians to learn to think abstractly—to free their imaginations from the tangible. And to do this, first they had to learn to think symbolically—to move from concrete arithmetic and geometry into the world of abstract, algebraic symbolism. So, if you ever struggled with algebra, take heart: this long history shows that generations of famous mathematicians have done so, too.

学习符号思考

LEARNING TO THINK SYMBOLICALLY

自从近四千年前有记载以来,代数就一直是数学的一部分,但并不总是以我们今天学习的符号形式出现。事实上,在这四千年的大部分时间里,代数都是用文字和数字来书写的——尽管欧几里得著名的公元前 300 年教科书《几何原本》也包括几何图表,以帮助证明毕达哥拉斯定理之类的东西,并展示如何展开我们今天写成 ( a + b ) 2 的正方形。因此,“代数”是通过繁琐的文字问题或日益复杂的图表来传达的——尽管几何确实有其优势。例如,它是证明毕达哥拉斯定理最简单的方法。在图 1.1中,我给出了这种证明的代数改编,尽管古人只是重新排列了图表,以直观地显示阴影面积等于三角形相邻边正方形面积之和——这是一种非常聪明的方法!5

Algebra has been part of mathematics since records began nearly four thousand years ago, but not always in the symbolic form we learn today. In fact, for most of those four millennia it was written entirely in words and numerals—although works such as Euclid’s famous 300 BCE textbook Elements also included geometric diagrams, to help prove such things as Pythagoras’s theorem, and to show how to expand squares that we would write today as (a + b)2. So “algebra” was communicated in cumbersome word problems or increasingly complicated diagrams—although geometry did have its advantages. For instance, it’s the easiest way to prove Pythagoras’s theorem. In figure 1.1, I’ve given an algebraic adaptation of such a proof, although the ancients simply rearranged the diagram to show visually that the shaded area is equal to the sum of the areas of the squares on the adjacent sides of the triangle—a pretty clever approach!5

代数花了很长时间才从算术和几何中分离出来,成为一门独立的学科。直到中世纪,它才有了名字。时代,这要归功于九世纪波斯数学家穆罕默德·伊本穆萨 (al-)Khwārizmī。(阿拉伯语前缀 al- 不是 Khwārizmī 波斯名字的一部分,但由于他在巴格达工作,因此他以 al-Khwārizmī 这个名字为人所知。)他在哈里发马蒙的开创性巴格达大学或“智慧之家”学习,当时伟大的阿拉伯语翻译运动正处于鼎盛时期:希腊语、印度语和其他古代手稿从蓬勃发展的伊斯兰帝国的各个角落收集起来并翻译成阿拉伯语。帝国主义很少是道德的,而且往往是暴力的,但它最终会导致文化的相互影响,在这种情况下,富有远见的翻译运动非常重要,以至于到了 12 世纪,欧洲人开始学习阿拉伯语,以便将这些手稿翻译成拉丁语——包括托勒密的《天文学大成》和欧几里得的《几何原本》 ,以及新的阿拉伯语作品,如花拉子米的作品。“代数”这个名字来自他的书《Al-Jabr wa'l muqābalah》标题的第一个词——意思是类似《完成和平衡计算简明书》6

It took a long time for algebra to emerge from arithmetic and geometry as a separate subject. It didn’t even get its name until medieval times, and that was thanks to the ninth-century Persian mathematician Mohammed ibn-Mūsā (al-)Khwārizmī. (The Arabic prefix al- was not part of Khwārizmī’s Persian name, but since he worked in Baghdad, al-Khwārizmī is the name by which he became known.) He studied at Caliph al-Ma’mūn’s pioneering Baghdad-based university, or “House of Wisdom,” when the great Arabic translation movement was at its height: Greek, Indian, and other ancient manuscripts were being collected from all corners of the burgeoning Islamic empire and translated into Arabic. Imperialism is rarely ethical and often violent, but it can ultimately lead to cultural cross-fertilisation, and in this case the visionary translation movement was so important that by the twelfth century, Europeans were learning Arabic in order to translate these manuscripts into Latin—including Ptolemy’s Almagest and Euclid’s Elements, along with new Arabic works such as those of al-Khwārizmī. The name “algebra” famously comes from the first word in the title of his book Al-Jabr wa’l muqābalah—which means something like “The Compendious Book on Calculation by Completion and Balancing.”6

图像

图 1.1。证明毕达哥拉斯定理。所需直角三角形的四个副本构成了大正方形的四个角。三角形的斜边是c ,因此,通过减去四个三角形的总面积(即 4 倍)可以得出阴影正方形的面积c 2 。一个b2)从较大正方形(边长为a + b )的面积得出:c2=一个+b24一个b2=一个2+b2

FIGURE 1.1. Proving Pythagoras’s theorem. Four copies of the required right-angled triangle make up the four corners of the larger square. The hypotenuse of the triangle is c, so you find the area, c2, of the shaded square by subtracting the total area of the four triangles (that is, 4 times ab2) from the area of the larger square (whose sides are of length a + b): c2=a+b24ab2=a2+b2

从花拉子米所附的问题来看,他所说的“完成”的一个例子是“完成平方”,这是你在学校里可能学到的解决二次方程的方法。这类方程在汉密尔顿虚数的故事中发挥了作用。例如,考虑 x 2 + 1 = 0。今天的学生可以立即写下解:x = ± i,其中i代表“虚”数17法国人勒内·笛卡尔在十七世纪将“虚数”一词引入数学。他指出,“对于我们能想象的任何方程”,都有可能找到解,但“在许多情况下,并不存在与我们想象的相对应的量。”换句话说,他是说你可以“想象”负数平方根的解,但它们并不存在——至少不是传统意义上的数字。8

Judging from the problems al-Khwārizmī included, an example of what he meant by “Completion” is “completing the square,” the method you might have learned in school to solve quadratic equations. Such equations play a part in the story of Hamilton’s imaginary numbers. For example, consider x2 + 1 = 0. Today’s students can write down the solution immediately: x = ±i, where i stands for the “imaginary” number 1.7 Frenchman René Descartes introduced the term “imaginary” to mathematics in the seventeenth century. He noted that “for any equation we can imagine” it is possible to find solutions, and yet, “in many cases no quantity exists that corresponds to what one imagines.” In other words, he was saying that you can “imagine” solutions with square roots of negative numbers, but they don’t really exist—at least, not as numbers in the traditional sense.8

如果笛卡尔认为这样的数字不存在,那么八百年前,在花拉子密的时代,-1 的平方根被认为是“不可能的”,而得出它的方程式被认为是无解的,这也就不足为奇了。因此,和大多数古人一样,花拉子密只关注有正解的二次方程,因为即使是负数似乎也没有实际意义。(七世纪的印度数学家婆罗摩笈多远远领先于他的时代,因为他考虑了正解和负解。)

If Descartes thought such numbers didn’t exist, then it’s not surprising that eight hundred years earlier, in al-Khwārizmī’s day, the square root of −1 was considered “impossible,” and the equations that gave rise to it were deemed unsolvable. So, like most of the ancients, al-Khwārizmī focussed only on quadratic equations with positive solutions, for even negative numbers seemed to have no practical meaning. (The seventhcentury Indian mathematician Brahmagupta was way ahead of his time, for he considered both positive and negative solutions.)

花拉子米也没有用我们今天使用的符号形式写方程式。事实上,在现代人看来,他的书更多的是算术而不是代数,当它被翻译成拉丁文时,它在欧洲产生的一个重要影响是普及了印度-阿拉伯十进制计数系统,该系统最终演变成我们现代的计数系统。然而,花拉子米经常被称为“代数之父”。他可能使用了文字而不是符号,他所包含的问题可能很简单——他告诉我们,他的目的是教学生如何解决“继承、遗产、分割、诉讼和贸易案件以及他们彼此之间的所有交易,或土地测量、运河挖掘、几何计算和其他对象”中的基本问题涉及各种类型和种类。”但他系统地列出了文字形式的线性和二次方程,以及解决这些方程的算法方法——即找到“未知数”,即我们现代的xy。事实上,英文单词“算法”——意思是一组用于执行计算或其他操作的规则——来自“algorismi”,这是 al-Khwārizmī 的早期拉丁化尝试。9

Al-Khwārizmī didn’t write equations in the symbolic form we use today, either. In fact, to modern eyes his book is more arithmetical than algebraic, and one of its important impacts in Europe, when it was translated into Latin, was the popularisation of the Hindu-Arabic decimal system of numeration that eventually evolved into our modern one. Yet Al-Khwārizmī is often called the “father of algebra.” He may have used words rather than symbols, and the problems he included may have been simple—his purpose, he tells us, was to teach students how to solve basic problems in “cases of inheritance, legacies, partitions, lawsuits and trade, and in all their dealings with one another, or where the measuring of lands, the digging of canals, geometrical computation, and other objects of various sorts and kinds are concerned.” But he systematically set out word-form linear and quadratic equations, with algorithmic methods for solving them—that is, for finding the “unknown numbers,” our modern x’s and y’s. In fact, the English word “algorithm”—meaning a set of rules for performing a calculation or other operation—comes from “algorismi,” an early Latinised attempt at al-Khwārizmī.9

“代数之父”这样的称号让我不禁想问,是否也有一位“母亲”?花拉子米并非凭空而来——甚至连牛顿、麦克斯韦或爱因斯坦都不是凭空而来。花拉子米的大多数前辈都已湮没于历史长河之中,但或许其中也曾有女性参与其中。事实上,神秘的希帕蒂娅可能是我们所知的最接近“母亲”的人,尽管我们无法得知她有多大的独创性,因为她的作品只留下了碎片。但当时的信件表明,她确实写过一篇学术评论,评论了另一位“代数之父”称号的竞争者,即三世纪讲希腊语的亚历山大·丢番图,丢番图在解决代数应用题时使用单词缩写,在符号的发展过程中迈出了重要的一步。

Such an epithet as “father of algebra” makes me want to ask, was there a “mother,” too? Al-Khwārizmī didn’t arise in a vacuum—not even Newton or Maxwell or Einstein did. Most of al-Khwārizmī’s predecessors have been lost to history, but perhaps there were women involved somewhere along the line. As it is, the mysterious Hypatia may be the closest we have to a known “mother,” although the extent of her originality is not known, for only fragments of her work remain. But letters from the time show that she did write a learned commentary on another contender for the title of “father of algebra,” the third-century Greek-speaking Alexandrian Diophantus, who took a significant step in the process of developing symbols when he used word abbreviations in setting out algebraic word problems.

当然,在现代,我们也有一些优秀的女性代数学家,包括西澳大利亚大学的名誉教授谢丽尔·普雷格,她是许多年轻女性的当代数学榜样。再往前追溯一点,有开创性的埃米·诺特,爱因斯坦的同事。她被称为“现代代数之母”,因为她在现代代数概念方面的工作远远超出了我在这里谈论的学校水平的代数。半个世纪前——再举一位数学先驱女性——1872 年,玛丽·萨默维尔在去世前一天还在研究汉密尔顿的四元数,当时她将近 92 岁。她曾是数学天文学最新发展的著名专家,被她的同时代人称为“科学女王”。近年来,她再次受到人们的赞誉,包括——通过民众投票,击败同为苏格兰人的马克斯韦尔——成为 2017 年苏格兰皇家银行发行的十英镑聚合物钞票的封面人物。

In modern times, of course, we’ve had some fine female algebraists, including the University of Western Australia’s Emeritus Professor Cheryl Praeger, who has been a contemporary mathematical role model for many young women. Moving back in time a little, there’s the groundbreaking Emmy Noether, a colleague of Einstein. She has been dubbed the “mother of modern algebra,” for her work on modern algebraic concepts that go way beyond the school-level algebra I’m talking about here. Half a century earlier—to take just one more pioneering mathematical woman—in 1872 Mary Somerville had been studying Hamilton’s quaternions the day before she died, at the age of nearly ninety-two. She had been a famous expert on the latest developments in mathematical astronomy, and was known to her contemporaries as the “Queen of Science.” She has become celebrated again in recent times, including—by popular vote, and beating fellow Scot Maxwell for the honour—as the face of the polymer ten-pound note issued by Royal Bank of Scotland in 2017.

说到更遥远的过去,我最喜欢的“代数之父”是神秘的伊丽莎白时代数学家托马斯·哈里奥特。我并不孤单:英国代数学家詹姆斯·约瑟夫·西尔维斯特在 1883 年写给阿瑟·凯莱(汉密尔顿四元数的早期崇拜者)的一封信中,将哈里奥特描述为“现代代数之父”。哈里奥特也受益于早期数学家,但他死后出版的分析艺术实践》是第一本完全以符号形式编写方程式的代数教科书,基本上使用了现代符号。它出版于 1631 年,比花拉子密晚了大约八个世纪,比丢番图晚了十四个世纪。这表明数学家花了多长时间才学会符号化思考。

As far as the more distant past goes, my favourite “father of algebra” is the enigmatic Elizabethan mathematician Thomas Harriot. I’m in good company: in an 1883 letter to Arthur Cayley—an early admirer of Hamilton’s quaternions—the British algebraist James Joseph Sylvester described Harriot as “the father of current algebra.” Harriot, too, was indebted to earlier mathematicians, but his posthumous Artis Analyticae Praxis (Practice of the Analytical Art) is the first algebra textbook where the equations are written entirely symbolically, using essentially modern symbols. It was published in 1631, some eight centuries after al-Khwārizmī and fourteen centuries after Diophantus. That’s an indication of how long it took for mathematicians to learn to think symbolically.

哈里奥特本人没有发表任何数学和实验成果:他的第一位赞助人是风趣而又备受争议的沃尔特·雷利爵士,而他则忙于航行公海、躲避异教徒猎人,最重要的是,开发新的数学应用来协助雷利的航海事业。(在伊丽莎白时代,拼写尚未固定,以沃尔特爵士命名的美国小镇被拼写为 Raleigh。他本人也使用这种拼写,但他最常用的是 Ralegh,所以这是今天许多学者更喜欢的拼写。)

Harriot himself didn’t publish any of his mathematical and experimental work: with the colourful and controversial Sir Walter Ralegh as his first patron, he was too busy sailing the high seas, dodging heretic hunters, and, above all, developing new mathematical applications to aid Ralegh’s navigational enterprises. (In Elizabethan times, spelling was not yet fixed, and the American town named after Sir Walter is spelled Raleigh. The man himself used this spelling, too, but most often he used Ralegh, so this is the spelling many scholars prefer today.)

尽管哈里奥特从未发表过他的发现,但他留下了数千页手写的计算和观察记录——他在收集的大量数据和方程式中寻找隐藏的模式和普遍性时不断思考。这确实是代数的一个关键点:模式和普遍性。符号方程的美妙之处在于,当你一眼就能看到问题时,就会更容易看到这些普遍模式。比较一下:

Although Harriot never got around to publishing his discoveries, he left behind thousands of pages of handwritten calculations and observations— his thoughts in progress as he searched for hidden patterns and generalities in the vast amounts of data and equations he’d collected. And that is, indeed, a key point of algebra: pattern and generality. The beauty of symbolic equations is that it’s much easier to see these general patterns when you can see a problem at a glance. Compare this:

取未知数的平方,然后将未知数与其自身相加,并从平方中减去和;现在总数为八。

Take the square of the unknown number, then add the unknown number to itself and take the sum away from the square; now let the total be eight.

替换为:

with this:

x 2 – 2 x = 8。

x2 – 2x = 8.

不仅如此,最早的数学家们分别求解每个方程,但如果你能看到适用于方程x 2 – 2 x = 8 的任何方法也适用于任何相同形式的方程x 2ax = b ,那么就更容易了。最终,古代数学家们开始认识到这一点,但进展相对较慢,因为他们必须把所有这些模式记在脑子里,或者记在长而复杂的句子里,很容易忘记。

And there’s more: the earliest mathematicians solved each equation separately, but it’s easier if you can see that whatever method works for the equation x2 – 2x = 8 will also work for any equation of the same form, x2ax = b. Eventually, ancient mathematicians did begin to recognise this, but progress was relatively slow because they had to keep all these patterns in their heads, or in long, convoluted sentences, and it was easy to lose track.

第一个以透明、可识别的现代符号形式发表方程式的人是哈里奥特 (Harriot) 的遗嘱执行人,他们于 1631 年发表了方程式,然后是笛卡尔 (Descartes),他于 1637 年在《方法论》的附录中发表方程式。10 (之前也有过几次尝试,但符号系统——更恰当地说是缩写——很扭曲,也很特殊。)11甚至我们认为理所当然的 +、−、= 和 × 符号也只是在 17 世纪才开始广泛使用。这意味着,我们所知道的早期代数学家——古代美索不达米亚人、埃及人、中国人和希腊人、中世纪的印度人、波斯人和阿拉伯人,以及近代欧洲人——都主要用文字或图形来表达他们的方程式。

The first to publish any equation in a transparent, recognisably modern symbolic form were Harriot’s executors in 1631, and then Descartes in an appendix to his 1637 Discourse on Method.10 (There were a few earlier attempts, but the symbolism—more properly called abbreviation—was tortured and idiosyncratic.)11 Even the +, −, =, and × signs we take for granted only came into widespread use in the seventeenth century. Which means that the earlier algebraists we know of—the ancient Mesopotamians, Egyptians, Chinese, and Greeks, the medieval Indians, Persians, and Arabs, as well as the early modern Europeans—all had expressed their equations mostly in words or pictorial word images.

• • •

• • •

符号思维是一种独特的技能,这一点从漫长的历史中可以看出来。以我上面给出的单词问题为例:它是算法思维的一个例子。但符号思维不仅仅是算法思维,因为它的符号有时包含着一种新创造力的种子——一种新的深远而又经济的思维。

It is a singular skill to think symbolically, as this long history shows. Take the word problem I gave above: it is an example of algorithmic thinking. But symbolic thinking is algorithmic and more, for its symbols sometimes contain the seeds of a new kind of creativity—a new kind of far-reaching yet economical thought.

一个经典的例子是爱因斯坦的E = mc 2。爱因斯坦并没有着手寻找能量和物质之间的联系。相反,他只是想根据他的新相对论计算运动电子的动能,以便可以通过实验检验他的理论预测。然而几个月后,26 岁的爱因斯坦开始意识到这个方程的重要性。他在 1905 年的第五篇开创性论文中写了这个方程,这是他的奇迹之年,但他又花了两年时间才梳理出这个符号关系的完整而戏剧性的含义。意识到这不仅仅是对特定形式的能量和特定类型的物质的计算,它是普遍的:如果一个物体获得(或失去)能量,也会获得(或失去)质量。这个奇怪的想法与我们所有的常识经验格格不入——但它就在那里,隐藏在他的方程符号中。实验物理学家花了几十年的时间才通过实验证实了这个惊人的数学预测。12

A classic case is Albert Einstein’s E = mc2. Einstein did not set out to find the connection between energy and matter. Rather, he simply wanted to calculate the kinetic energy of a moving electron according to his new theory of relativity, so that his theoretical prediction could be tested experimentally. A few months later, however, twenty-six-year-old Einstein began to realise the significance of his equation. He wrote it up in his fifth groundbreaking paper of 1905, his annus mirabilis, but it would take him two more years to tease out the full, dramatic implications of this symbolic relationship. To realise that this wasn’t just a calculation about a particular form of energy and a particular type of matter, it was general: if a body gains (or loses) energy, it also gains (or loses) mass. This bizarre idea is alien to all our commonsense experience—but there it was, hidden in the symbols of his equation. It took experimental physicists decades to experimentally confirm this astonishing mathematical prediction.12

一个更简单、更早的例子是幂序列x、x 2x 3等。第一个“幂”是 1,所以x实际上是x 1,其中 1 传统上在几何上与一维线相关。接下来的两个幂x 2x 3发音为“ x平方”和“ x立方”,类比正方形的面积和立方体的体积。这些名称突显了早期数学家以几何方式而非代数方式思考的方式,因为几何具有有形的性质。相比之下,符号代数是抽象的:你必须赋予它意义,即使它只是显示一个有趣的模式,例如x、x 2x 3x 4 ……但这种灵活性是代数的强大之处。你可以写下任意数量的(有限)高次幂,而不必将它们可视化为物理对象。

A much simpler and earlier example is the sequence of powers x, x2, x3, and so on. The first “power” is 1, so x is really x1, where the 1 was traditionally linked geometrically to a 1-D line. The next two, x2 and x3, are pronounced “x squared” and “x cubed” by analogy with the area of a square and the volume of a cube. These names highlight the way that early mathematicians thought geometrically rather than algebraically, because of the tangible nature of geometry. By contrast, symbolic algebra is abstract: you have to give it meaning, even if it is simply the display of an interesting pattern such as x, x2, x3, x4, … But this flexibility is algebra’s great strength. You can write down as many (finite) higher powers as you like, without having to visualise them as physical objects.

今天看来,这似乎显而易见,但数学家们花了三千五百年的时间才从解决二次方程(“二次”源于拉丁语“平方”,因此二次方程的最高幂是x 2(古人称之为未知数乘以自身)——到解决“三次”和更高次方程。当然,这些高次方程要困难得多;但解法不易的部分原因是代数长期以来与文字和具体图像紧密相关。

This may sound obvious today, but it took three and a half thousand years for mathematicians to move from solving quadratic equations— “quadratic” derives from the Latin for “square,” so quadratic equations are those whose highest power is x2 (the unknown multiplied by itself, as the ancients put it)—to solving “cubic” and higher equations. These higherdegree equations are much more difficult, of course; but part of the reason solutions didn’t come easily was that algebra was tied to words and concrete images for such a very long time.

例如,我提到了花拉子米为了解决二次方程而采用的“配方法”。这实际上是一个有四千年历史的问题,可以追溯到(就历史记录而言)由生活在现代伊拉克地区的数学家制作的楔形文字板。这些古代美索不达米亚人通过配方法来解决二次方程。这是当时一个典型的教学问题:“将我的长度的 20 倍加到我的正方形面积上,[得到] 21。我的正方形有多方?” 13这类问题及其解决方法与今天教授的算法相似——只是四千年前,这种方法完全是几何学方法。首先,画一个正方形任意边长为x(现代符号);然后向其中添加一个尺寸为 20 × x的矩形。现在将这个额外的矩形分成两个相等的小矩形,并将它们排列在原始正方形的旁边和下面。最后,完成这个新的更大的正方形,如图1.2所示。

For instance, I mentioned al-Khwārizmī’s “completing the square” in order to solve a quadratic equation. It’s actually a four-thousand-year-old problem, dating back (as far as the historical record shows) to cuneiform tablets made by mathematicians living, like al-Khwārizmī, in the region of modern-day Iraq. These ancient Mesopotamians solved quadratic equations by literally completing a square. Here is a typical teaching problem of the time: “Add 20 of my length to the area of my square, [to get] 21. How square is my square?”13 This type of problem, and the algorithm for solving it, is similar to those taught today—except that four millennia ago, the method was worked out entirely geometrically. First, draw a square of arbitrary side x (in modern notation); then add to it a rectangle of dimensions 20 × x. Now split this additional rectangle into two equal smaller ones and arrange them beside and below the original square. Finally, complete this new, larger square, as in figure 1.2.

图像

图 1.2。在现代代数符号中,这个古老的问题是方程式x 2 + 20 x = 21 的文字形式。与其古老的几何对应物一样,完成平方的代数算法需要记住在两边加上“(x的系数的一半)平方”的步骤,这样等式就变成了 ( x + 10) 2 = 121。然后,原始正方形所需的边长为=12110=1

FIGURE 1.2. In modern algebraic symbolism, this ancient problem is the word form of the equation x2 + 20x = 21. Like its ancient geometric counterpart, the algebraic algorithm for completing the square involves memorising the step of adding “(half the coefficient of x) squared” to both sides, so that the equation becomes (x + 10)2 = 121. Then the required side of the original square is x=12110=1.

美索不达米亚人在开发这种方法时,至少在最初阶段,考虑到了实际问题。他们生活在水资源极为稀缺的地区,他们的石碑上记载了许多与运河和水库开挖、蓄水池容量、水坝和堤坝的修建以及与这些任务相关的行政账目有关的问题——为了解决这些问题,这些古代数学家必须解出与面积和体积有关的方程式。14近三千年后,花拉子密也关注类似的实际问题,他使用了类似的几何方法来完成平方——直到十七世纪,其他数学家也是如此。

The Mesopotamians had practical problems in mind when they developed this method, at least initially. Living in a land where water was at a premium, their tablets contain many problems relating to canal and reservoir excavations, the capacity of cisterns, the construction and repair of dams and levees, and administrative accounts relating to these tasks—and to solve these problems, these ancient mathematicians had to solve equations relating to areas and volumes.14 Nearly three thousand years later, al-Khwārizmī, too, focussed on similar practical problems, and he used a similar geometrical method of completing the square—and so did other mathematicians right up to the seventeenth century.

伊斯兰数学家沙拉夫丁·塔西是最早在寻找三次方程解方面取得进展的人之一,大约在公元 1200 年,但第一个发表正确的通用三次算法的人是意大利数学家吉罗拉莫·卡尔达诺,发表在他 1545 年出版的《大术》一书中。15他之前的所有人一样,他仍然用文字(或文字缩写)而不是符号来写出他的解,并且他仍然以几何方式设计他的方法——通过令人惊叹的可视化壮举真正完成一个立方体。

The Islamic mathematician Sharaf al-Dīn al-Ṭūsī was one of the earliest to make progress in the search for solutions of cubic equations, in about 1200 CE, but the first to publish correct general cubic algorithms was the Italian mathematician Girolamo Cardano, in his 1545 book Ars Magna (The Great Art).15 Like everyone before him, he still wrote his solutions in words (or abbreviations of words) rather than symbols, and he still devised his method geometrically—literally completing a cube in a stunning feat of visualisation.

一场数学决斗、一个棘手的方程和一个虚数

A MATHEMATICAL DUEL, A PESKY EQUATION, AND AN IMAGINARY NUMBER

卡尔达诺不仅是一位天才数学家,还是一位医生、占星家、赌徒、哲学家和神秘主义者,他相信自己最好的想法来自夜晚拜访他的灵魂。然而,在三次方程的问题上,他的灵感来自他的同胞尼科洛·塔尔塔利亚,而不是他忠实的灵媒。卡尔达诺听说塔尔塔利亚破解了这个问题,他非常好奇,于是缠着他透露他的方法——他甚至提出利用自己的关系,让身无分文的塔尔塔利亚与有影响力的人取得联系,这些人可能会为他在弹道学等有用主题上的工作付钱。塔尔​​塔利亚最终让步,条件是卡尔达诺对该方法保密——塔尔塔利亚自然想自己出版它,或者更好的是,把它提供给未来的赞助人。

As well as being a talented mathematician, Cardano was a physician, an astrologer, a gambler, something of a philosopher, and a mystic who believed that his best ideas came from a spirit who visited him at night. In the case of cubic equations, however, he received his inspiration from his countryman Niccolò Tartaglia rather than his faithful ethereal advisor. Cardano had heard that Tartaglia had cracked the problem, and he was so intrigued that he badgered him to reveal his method—he even offered to use his connections to put the impecunious Tartaglia in touch with influential people who might pay him for his work on such useful topics as ballistics. Tartaglia finally relented, on condition that Cardano keep the method secret—Tartaglia naturally wanted to publish it himself or, better still, offer it to a future patron.

几年后,当塔塔利亚仍保守秘密时,卡尔达诺发现西皮奥内·德尔·费罗也比塔塔利亚早找到了解决方案。因此,卡尔达诺觉得他可以违背诺言,发表论文——他总是在寻找宣传机会。但他完全承认这两个人,并超越他们,解决了更广泛、更普遍的方程式。尽管如此,塔塔利亚还是很生气——以至于他向卡尔达诺发起公开决斗,不是用剑,而是用解题比赛。卡尔达诺谨慎地拒绝了:在这种竞争激烈的文艺复兴盛会中,名誉(和工作)很容易赢得和失去。此外,塔塔利亚已经与德尔·费罗的学生安东尼奥·菲奥尔较量过,后者知道他老师的三次方法——而塔塔利亚赢得了那场比赛。

Some years later, while Tartaglia was still holding onto his secret, Cardano discovered that Scipione del Ferro had also found the solution, before Tartaglia. So, Cardano felt he could break his promise and publish—he always had his eye on a publicity opportunity. But he fully acknowledged both men, and he went beyond them in solving a wider, more general range of equations. Still, Tartaglia was furious—so much so that he challenged Cardano to a public duel, not with swords but with a problem-solving competition. Cardano prudently refused: reputations (and jobs) were easily won and lost in these fiercely competitive Renaissance spectacles. Besides, Tartaglia had already taken on del Ferro’s student Antonio Fior, who knew of his teacher’s cubic method—and Tartaglia had won that match.

在书中,卡尔达诺用一页巧妙的几何类比解释了他的通用算法,然后给出了具体的说明性例子。他是这样解释他求解x 3 = 6 x + 40 的方法的(使用现代符号,我也将使用现代符号使卡尔达诺的算法更容易理解;请耐心听我说,即使你只是浏览一下,因为最后一行的表达式形式与虚数和向量的故事有着惊人的相关性):“将 2(x的系数的三分之一)的立方乘以 8;用 400(20 的平方)减去这个数字,[这是常数的一半,等于 392;它的平方根加上 20 等于20+392并从 20 个品牌中减去20392;这些数的立方根之和,20+3923+203923,是x的值。”呼!你不得不佩服他的耐心,想出了这么复杂的问题。16

In his book, Cardano explained his general algorithm in a page of ingenious geometrical analogy and then gave specific illustrative examples. This is how he explained his method for solving x3 = 6x + 40 (to use modern notation, which I’ll also use to make Cardano’s algorithm a little easier to follow; bear with me, even if you just skim through it, because the form of the expression in the last line has surprising relevance to the story of imaginary numbers and vectors): “Raise 2, one-third the coefficient of x, to the cube, which makes 8; subtract this from 400, the square of 20, [which is] one-half of the constant, making 392; the square root of this added to 20 makes 20+392, and subtracted from 20 makes 20392; and the sum of the cube roots of these, 20+3923+203923, is the value of x.” Phew! You’ve got to admire his patience in coming up with something so convoluted.16

从向量发展史和数学发展史的角度来看,有趣的是,当这种解的平方根号下的数字为负时会发生什么。也就是说,当你有一个虚数,例如121

The interesting thing, from the point of view of the story of vectors— and of the development of mathematics in general—is what happens when the number under the square root sign in such a solution is negative. That is, when you have an imaginary number such as 121.

美索不达米亚人忽略了二次方程的负解和虚解,因为它们与他们试图解决的实际问题无关——田地和运河的维度不能为负数或虚数。希腊人也是如此,直到花拉子米和塔西,直到卡尔达诺被迫与这些“不可能”的数字搏斗。他研究方程数学只是为了它本身,为了智力挑战——他被这样一个事实所困扰:如果他用他用于x 3 = 6 x + 40 的相同方法将其应用于x 3 = 15 x + 4,那么x的值将是

The Mesopotamians had ignored negative and imaginary solutions of quadratic equations because they had no relevance to the practical problems they were trying to solve—you can’t have negative or imaginary dimensions of fields and canals. Similarly for the Greeks, through to al-Khwārizmī and al-Ṭūsī, and right up until Cardano was forced to wrestle with these “impossible” numbers. He was studying the mathematics of equations simply for its own sake, for the intellectual challenge of it—and he was flummoxed by the fact that if he took the same method he’d used for x3 = 6x + 40 and applied it to x3 = 15x + 4, then the value of x turned out to be

2+1213+21213

2+1213+21213.

卡尔达诺的结论是,这样的解决方案是“复杂的”,“既微妙又无用”——因为除了不受欢迎的121,他已经知道事实上x = 4。他知道这一点是因为他会通过猜测解决方案来开始理解问题——数学家们一直在做的事情。当没有已知的解决问题的算法时,这种方法尤其有用,所以它是古代代数的开始。对于卡尔达诺方程x 3 = 15 x + 4,你可以通过尝试一个简单的可能值(例如x = 3)来了解这个想法;比较每一边你会发现这个值太小了,所以试试x = 4。在这种情况下它马上就起作用了,但有时你必须尝试中间值。这仍然是数学家“数字化”解决难题的方式,尽管他们有算法(现在还有计算机)来有效和详尽地选择他们的猜测。

Cardano concluded that such a solution was “sophistic,” and “as subtle as it is useless”—because aside from the unwelcome 121, he already knew that in fact x = 4. He knew this because he would have begun to understand the problem by guessing the solution—something mathematicians have always done. It is especially useful when there isn’t a known algorithm for solving a problem, so it is the way ancient algebra began. For Cardano’s equation x3 = 15x + 4, you can see the idea by trying a simple possible value such as x = 3; comparing each side you see that this is too small, so try x = 4. In this case it works straightaway, but sometimes you have to try intermediate values. This is still the way mathematicians solve difficult problems “numerically,” although they have algorithms (and now computers) to choose their guesses efficiently and exhaustively.

十五年后,大约 1560 年,又出现了另一个优秀的早期现代意大利代数学家拉斐尔·邦贝利(Rafael Bombelli)重新审视了卡尔达诺的难题。问题是,=2+1213+21213考虑到解是x = 4,这和它有什么关系?经过一番思考后,Bombelli 突然有了所谓的“疯狂想法”:如果你能将121像这样:121×1, 要得到111那么,你能找到现在所谓的“复数”吗?复数是实数和虚数的混合,它的立方是2+111令人惊讶的是,经过反复尝试和耐心等待,他发现2+12+1113,你可以看到如果你乘以2+13。同样,他发现2121113按照 Cardano 的解决方案将它们加在一起,你会得到

Fifteen years later, around 1560, yet another excellent early modern Italian algebraist, Rafael Bombelli, took another look at Cardano’s conundrum. The question was, what did x=2+1213+21213 have to do with it, given that the solution was x = 4? After a great deal of thinking Bombelli suddenly had what he called “a wild thought”: what if you could factor 121 like this: 121×1, to get 111? And could you then find what is nowadays called a “complex” number—a mix of real and imaginary numbers—whose cube is 2+111? Amazingly, with trial, error, and a lot of patience he found that 2+1 is a solution of 2+1113, as you can see if you multiply out 2+13. Similarly, he found that 21 is a solution of 21113. Adding these together as in Cardano’s solution, you get

=2+1+21

x=2+1+21,

看似奇迹般地,得出了x = 4。谜团解开了!

which, seemingly miraculously, gives x = 4. Mystery solved!

不过,它只针对这个特殊情况得到了解决,当邦贝利事先知道x = 4 时——他对操纵虚数有着绝妙的洞察力,但他没有通用算法。他也没有用我在这里给出的透明现代符号形式来写他的方程式——而且像卡尔达诺一样,他贬低1他认为这个奇怪的数字“诡辩”。但当他的著作《代数学》于 1570 年代出版时,他确实将这个奇怪的数字更坚定地放在了数学雷达上。当时,他或其他任何人都不知道它会变得多么有用——在汉密尔顿的四元数和向量中,以及在工程、计算和量子理论等领域。17

It was solved only for this special case, though, and when Bombelli knew beforehand that x = 4—he’d had a brilliant insight about manipulating imaginary numbers, but he had no general algorithm. He didn’t write his equations in the transparent modern symbolic form I’ve given here, either—and like Cardano, he disparaged 1 as “sophistic.” But he did put this strange number more firmly on the mathematical radar when his book Algebra was published in the 1570s. Little did he or anyone else know back then just how useful it would become—in Hamilton’s quaternions and vectors, and in fields such as engineering, computing, and quantum theory.17

至于三次方程,哈里奥特于 1600 年左右首次发现了一般的符号代数解——而且他的证明中没有参考几何。约翰·沃利斯——也许是哈里奥特和牛顿时代之间最优秀的英国数学家——是少数几个承认哈里奥特将代数从几何中解放出来的成就的近现代人之一,用沃利斯的话来说,他“将代数视为纯粹的代数,从其自身的原理出发,不依赖于几何或与几何有任何联系。” 18这为我将要讨论的汉密尔顿奠定了一些数学背景,因为他也想找到一种纯代数表示几何运算的方法——在他看来,不是立方体而是空间旋转,这个问题的解将使他发明向量。

As for cubic equations, it was Harriot who first found general, symbolic algebraic solutions, sometime around 1600—and with no reference to geometry for his proofs. John Wallis—perhaps the best British mathematician between Harriot’s time and Newton’s—was one of the few nearcontemporaries to recognise Harriot’s achievement in liberating algebra from geometry, treating, as Wallis put it, “algebra purely by itself, and from its own principles, without dependence on geometry, or any connexion therewith.”18 This sets some mathematical context for where I’ll be going with Hamilton, for he, too, wanted to find a way to represent purely algebraically a geometrical operation—in his case, not cubes but rotations in space, the problem whose solution will lead him to invent vectors.

用代数来设想几何不仅扩展了代数,也扩展了几何,我们将看到,随着向量和张量的出现,这两种数学齐头并进,相互影响。但第一步是像哈里奥特和沃利斯那样,认识到代数本身就是一门学科,就像几何一样。

Using algebra to envisage geometry expands not just algebra but geometry, too, and we’ll see that these two kinds of maths went hand in hand, each influencing the other, as vectors and tensors emerged. But the first step had been to see, as Harriot and Wallis did, that algebra was a subject in its own right, just as geometry was.

哈里奥特的灵感来自多才多艺的早期现代法国人弗朗索瓦·韦达,后者开始使用大写字母表示未知数,哈里奥特刻苦钻研其关于三次方程的论文。哈里奥特像我们今天一样使用小写字母,他如此全面地使用符号,以至于他成为了符号思维大师。他的一个洞见是表明多项式方程可以通过乘以它们的因子来生成——例如,两个线性因子生成一个二次方程,三个线性因子生成一个三次方程,四个线性因子生成一个四次方程,等等。这个“因式定理”现在看起来可能很明显——你可能在高中代数课上学过它——但在哈里奥特之前没有人写过这样的符号方程

Harriot had taken his lead from the versatile early modern Frenchman François Viète, who had begun to use uppercase letters for unknowns, and whose treatise on cubic equations Harriot studied assiduously. Harriot used lowercase letters as we do today, and he used symbols so completely that he became a master of symbolic thinking. One of his insights was to show that polynomial equations can be generated by multiplying their factors—for example, two linear factors generate a quadratic, three give a cubic, four a quartic, and so on. This “factor theorem” may seem obvious now—you may have learned it in a senior high school algebra class—but no one before Harriot had written symbolic equations such as

n=0.

(xl)(xm)(xn)=0.

实际上,哈里奥特并没有使用单独的圆括号来表示乘积,而是将因数一个接一个地写在上方,并用方括号括住群。他倾向于使用a而不是x来表示未知数,使用aa而不是a 2 。x 、x 2x 3x 4 、...这些符号是笛卡尔于 1637 年发表的,尽管他有时仍像哈里奥特一样使用xx甚至aa。无论如何,这个方程暗示的是,三次方程必须有三个解,x = l、x = m、x = n,无论它们是正数还是负数,实数还是虚数。相比之下,卡尔达诺的算法提到“唯一”解时,好像只有一个——如果你用物质立方体来想象它,你就会期望它只有一个。19

Actually, Harriot didn’t use separate round brackets for products, but wrote the factors one on top of the other with a square bracket around the group. And he tended to use a rather than x for the unknown, and aa instead of a2. We owe the notation x, x2, x3, x4, … to Descartes, who published it in 1637, although he still sometimes used xx and even aa, like Harriot. Either way, what this equation hints at is that a cubic equation has to have three solutions, x = l, x = m, x = n, whether they are positive or negative, real or imaginary. By contrast, Cardano’s algorithm had spoken of “the” solution, as if there were only one—which is what you’d expect if you were imagining it in terms of a material cube.19

要了解哈里奥特符号体系的优势(它与我在这里使用的现代版本没有太大区别,并且同样透明),请考虑从邦贝利的x = 4 的出发点来求解那个棘手的卡尔达诺方程。哈里奥特的方法建议首先将x 3 = 15 x + 4 写为x 3 − 15 x − 4 = 0。这正是你在高中时会做的事情,然后将x 3 − 15 x − 4 除以x − 4 得到x 3 − 15 x − 4 =x − 4)(x 2 + 4 x + 1)。当x = 4x 2 + 4 x + 1 = 0时,该式等于零。你可以完成平方来求解二次方程,以找到另外两个解,=2+3=23,总共有三个解。在这种情况下,所有解都是实数,而 Cardano 的复杂表达式=2+1213+21213似乎没有。……然而,后来的数学家们通过发现复数本身实际上有三个立方根,将历史点联系起来。因此,卡尔达诺的棘手方程的三个实数解可以从他的算法中恢复出来!我在尾注中展示了如何做到这一点。20

To see the advantage of Harriot’s symbolism, which was not too different from the modern version I’m using here, and just as transparent, consider solving that pesky Cardano equation from Bombelli’s starting point of knowing that x = 4. Harriot’s method suggests that first you write x3 = 15x + 4 as x3 − 15x − 4 = 0. This is just what you would have done in high school, and you’d then divide x3 − 15x − 4 by x − 4 to get x3 − 15x − 4 = (x − 4)(x2 + 4x + 1). This equals zero when x = 4 or when x2 + 4x + 1 = 0. You can complete the square to solve the quadratic, to find two additional solutions, x=2+3 and x=23, making a total of three solutions. In this case, all the solutions turn out to be real, and Cardano’s complicated expression x=2+1213+21213 doesn’t come into it. Or so it seems.… Later mathematicians, however, would connect the historical dots by discovering that in fact complex numbers themselves each have three cube roots. So, the three real solutions of Cardano’s pesky equation can be recovered from his algorithm! I’ve shown how in the endnote.20

符号思考的力量

THE POWER OF THINKING SYMBOLICALLY

因子法在今天已经是基本方法了,但在四百年前,它可是一个巨大的突破。哈里奥特并不总是使用它,而且它的完整、更普遍的含义(代数基本定理)在两个世纪后才得到严格证明。因此,在韦达和卡尔达诺之后,他还设计了一整套算法,用于解决各种类型的二次、三次和四次方程。但他很清楚代数符号的价值。“当我们的这种归约法直接展示所有根(即所有解)时,就不需要冗长的规则了,”他说(因为就连韦达也很啰嗦),“不仅适用于这种类型的方程,也适用于任何其他情况。” 21

The factor approach is elementary today, but it was a huge breakthrough four hundred years ago. Harriot didn’t always use it, and its full, more general implication (the fundamental theorem of algebra) would not be proved rigorously for another two centuries. So, following Viète and Cardano, he also devised a whole list of algorithms for solving various types of quadratic, cubic, and quartic equations. But he was clear about the value of algebraic symbolism. “What need is there for verbose precepts,” he said (for even Viète was wordy), “when our kind of reduction exhibits all the roots [that is, all the solutions] directly, not only for this type of equation, but for any other case you like.”21

他想表达的是,用符号进行概括比用文字容易得多。当你能够概括时——当你能够看到适用于出乎意料的广泛问题的共同模式时——你就可以在科学技术和数学上取得非凡的进步。例如,麦克斯韦能够展示光的电磁波性质,并预测无线电波的存在,因为他对电磁学的数学分析得出了与描述拨动吉他或小提琴弦时看到的波形相同的方程。诺特巧妙地概括了对称性的数学模式与能量和动量等物理量守恒之间的关系。

What he was getting at is that generalisation is far easier in symbols than in words. And when you can generalise—when you can see common patterns that apply to an unexpectedly wide range of problems—you can make extraordinary progress in science and technology as well as mathematics. For instance, Maxwell was able to show the electromagnetic wave nature of light, and to predict the existence of radio waves, because his mathematical analysis of electromagnetism turned up the very same kind of equation that had been used to describe the wave pattern you see when you pluck a guitar or violin string. And Noether brilliantly generalised the relationship between mathematical patterns of symmetry and the conservation of physical quantities such as energy and momentum.

稍后将详细介绍这些例子;与此同时,哈里奥特学者穆里尔塞尔特曼巧妙地总结了哈里奥特的重要性和代数符号的力量:

More on these examples later; meantime, Harriot scholar Muriel Seltman sums up neatly both Harriot’s importance and the power of algebraic symbolism:

符号和数学思维过程之间存在着相互关系,很难高估哈里奥特的技巧和清晰的思维效果,这些技巧和清晰的思维通过符号来指导你的工作从而使数学以一种全新的方式变得易于理解……可视化是显而易见的,但非常重要。现在可以将符号当作不可可视化的概念来操纵,符号只是其具体化。22

There is a reciprocal relation between symbolism and mathematical thought-processes, and it would be hard to overestimate the effect of Harriot’s techniques and clarity of thought expressed in a symbolism that directs what you do visually and therefore makes mathematics accessible in a totally new way…. The visualizability is obvious but profoundly important. It is now possible to manipulate the symbol as if it were the non-visualizable concept of which the symbol is only the embodiment.22

这也让我想到了最终要说的。汉密尔顿的四维四元数没有传统的具体几何类似物,我们大多数人也不可能想象爱因斯坦的四维时空(更不用说十维弦理论了);但描述它们的数学符号方程却有自己的可触性。它们可以被操纵,就好像四维“真的”存在一样,因为从代数上讲,x4与“正方形” x2和“立方体” x3一样有效而且——跳到坐标和向量——(a ,b,c,d )与( a,b,c )是同一类东西。

Which brings me to where I’m ultimately heading. Hamilton’s fourdimensional quaternions don’t have traditional concrete geometrical analogues, and it’s impossible for most of us to visualise Einstein’s fourdimensional space-time (let alone such things as ten-dimensional string theory); but the mathematical symbolic equations that describe them have their own kind of tangibility. They can be manipulated as if four dimensions “really” do exist, because algebraically speaking x4 is just as valid as the “square” x2 and the “cube” x3, and—jumping ahead to coordinates and vectors—(a, b, c, d) is the same kind of thing as (a, b, c).

随着代数符号的兴起,一种新的抽象思维应运而生,为微积分的发展开辟了道路,并最终为矢量和张量微积分的发展开辟了道路。这种奇妙的语言使数学家能够解决大量新的、更复杂的问题,这些问题的解决方案不仅为我们带来了新技术,也为我们提供了看待现实的新方式。因此,下一章将简要介绍微积分的非凡历史。

With the rise of algebraic symbolism, a new kind of abstract thinking arose, which opened the way for the development of calculus—and ultimately to vector and tensor calculus. This marvelous language has enabled mathematicians to solve a vast range of new, more complicated problems—problems whose solutions have given us new technologies as well as new ways of seeing reality. So the next chapter will offer a brief tour through the extraordinary story of calculus.

(2)微积分的诞生

(2) THE ARRIVAL OF CALCULUS

在布鲁姆大桥获得“电”学发现的十年前,汉密尔顿利用微积分做出了他的第一个重大发现:一种新的光学现象的理论存在——某些晶体中存在一种特殊类型的折射,以前从未有人观察到过。

A decade before his “electric” insight at Broome Bridge, Hamilton had used differential calculus to make his first major discovery: the theoretical existence of a new optical phenomenon—a special type of refraction in certain crystals, which no one had ever observed before.

光学是研究光的行为的学科,尽管没有人知道光到底什么,但对于物理学家和数学家来说,光学都是一个令人兴奋的课题。然而,越来越多的物理学家相信,光不是牛顿所认为的粒子,而是某种。麦克斯韦将把这一结论推向最终结论——强调向量微积分在这一过程中的重要性——但潮流在 1801 年开始转变,当时有一项非凡的实验,如今已成为物理学史上的经典:托马斯·杨的“双缝”实验。

Optics, the study of how light behaves, was an exciting topic for both physicists and mathematicians, even though no one knew what light actually was. Yet a growing number of physicists were confident that it wasn’t a particle, as Newton had believed, but a wave of some sort. Maxwell will take this to its ultimate conclusion—highlighting the importance of vector calculus in the process—but the tide had begun to turn in 1801, with a remarkable experiment that is now a classic in the story of physics: Thomas Young’s “double slit” experiment.

这个想法是为了观察两束光相互作用时会发生什么,并看看由此产生的相互作用模式是波还是粒子的模式。如果你把两块鹅卵石扔进池塘,你就能看到波产生的模式。石头撞击水面时产生的涟漪以每块石头为中心向外扩散,形成同心圆——你可以观察一块石头的涟漪跳过另一块石头周围的涟漪的地方。当它们跳过时,水位上升,所以两个波​​纹会结合在一起形成一个更大的波浪。如果你能用光做同样的实验——如果光确实以波的形式传播——你会看到这种效果,即两波光波相互加强,亮度增加。更难看到的是,在水面的某些地方,波纹会相互抵消——一个波的波峰与另一个波的波谷相遇——所以水面是平的。这种效果的光学类似物是一个暗斑——没有光,因为波相互抵消了。但光是波吗?这就是杨开始探索的。

The idea was to observe what happens when two beams of light interact with each other, and to see if the resulting interaction patterns were those of waves or particles. You can see the pattern that waves produce if you drop two pebbles into a pond. The impact of the stones as they hit the water produces ripples that spread out in concentric circles centred around each of the stones—and you can watch the places where the ripples from one stone jump over those around the other. And when they do, the water level rises, so the two ripples combine to produce a bigger wave. If you could do the same experiment with light—and if it did travel as a wave—you’d see this effect as an increase in brightness as the two light waves reinforced each other. It’s harder to see that at some places on the water surface the ripples cancel each other out—the peak of one wave has met up with the trough of the other—and so the water surface is flat. The optical analogue of this effect would be a dark patch—there’d be no light because the waves canceled each other. But was light a wave? That is what Young set out to explore.

他通过两个小针孔照射光线,产生两束光,然后让两束光照射到屏幕上,从而呈现出“干涉图样”。如果光是由粒子组成的,那么每束光都会笔直向前,因此你会看到两个明亮的斑块与两个针孔成一线——类似于两堆信件被扔进有两个开口的邮箱,或者两束乒乓球射向屏幕。然而,如果光以波的形式传播,那么它会在针孔周围弯曲(或“衍射”),产生像池塘上的涟漪一样的圆形波——而交叉的涟漪会在屏幕上产生明暗斑块的干涉图样。果然,这正是杨所发现的。这是一个美丽的实验。然而,一个世纪后,爱因斯坦提出了光量子的概念,后来被称为光子。光子不是牛顿式的物质粒子,但在与原子的相互作用中,它们可以表现出粒子般的行为。除了这一奇怪的发现之外,量子物理学家还改编了杨的实验,表明光子电子等亚原子物质粒子都表现出波干涉图案。换句话说,光和物质都可以表现出波状或粒子状的行为,具体取决于情况。但这只是后见之明,所以让我们回到杨的光波动理论,以及汉密尔顿对它的贡献。

He created two beams of light by shining light through two tiny pinholes, and he manifested the resulting “interference pattern” by letting the two beams shine onto a screen. If light were made of particles, each beam would just go straight ahead, so you’d expect to see just two bright patches in line with the two pinholes—analogous to two piles of letters dropped into a postbox with two openings, or two streams of ping-pong balls fired at a screen. If light traveled as waves, however, then it would bend (or “diffract”) around the pinholes to produce circular waves like the ripples on the pond—and the crisscrossing ripples would produce an interference pattern of light and dark patches across the screen. And sure enough, this is just what Young found. It was a beautiful experiment. A century later, however, Einstein would introduce the idea of a light quantum, later called a photon. Photons are not Newtonian-style material particles, but in their interactions with atoms they can behave in a particle-like way. To add to this bizarre finding, quantum physicists have adapted Young’s experiment to show that both light photons and subatomic material particles such as electrons exhibit wave interference patterns. In other words, both light and matter can manifest wave-like or particle-like behavior, depending on the situation. But this is hindsight, so let’s return to Young’s wave theory of light, and what Hamilton did with it.

杨是玛丽·萨默维尔的朋友,她回忆说,起初英国很少有人认真对待杨的光波推论,因为没有人想贬低伟大的牛顿。(事实上,当时他有充分的理由拒绝波动理论。)但阻碍杨实验被接受的不仅仅是牛顿的名声:还有一个事实是,没有人知道光波可能是由什么构成的,或者它是如何从太阳到地球的旅程中穿越空旷的空间的。尽管如此,这并没有阻止 19 世纪早期的数学家,如汉密尔顿,从理论上探索光波——这是研究人员自牛顿时代的克里斯蒂安·惠更斯以来一直在做的事情。1

Young was a friend of Mary Somerville, and she recalled that few in England took his light-wave deduction seriously at first, because no one wanted to diminish the great Newton. (In fact, at the time he’d had good reasons for rejecting a wave theory.) But it wasn’t just Newton’s reputation that got in the way of the reception of Young’s experiment: there was also the fact that no one knew what a light wave could possibly be made of, or how it managed to travel through empty space in its journey from the Sun to Earth. Still, this hadn’t stopped early nineteenth-century mathematicians such as Hamilton from exploring light waves theoretically—something researchers had been doing since Christian Huygens in Newton’s time.1

汉密尔顿在对光波穿过各种形状的晶体时的行为方式进行新分析时发现,当这些波以特定方向进入某种类型的晶体(称为“双轴”)时,它们应该被折射成一个射线,而不是你在普通棱镜中看到的熟悉的带状。如今,激光束的“锥形折射”正在找到各种实际用途,包括作为操纵化学分子或血细胞的光镊,以及自由空间光通信。2

In his new analysis of the way light waves should behave when they pass through crystals of various shapes, Hamilton found that when these waves passed into a certain type of crystal (called “biaxial”) in a certain direction, they should be refracted into a cone of rays rather than the familiar band you see with an ordinary prism. Today, the “conical refraction” of laser beams is finding various practical uses, including as optical tweezers for manipulating chemical molecules or blood cells, and in free-space optical communication.2

然而,早在 1832 年初,还没有人见过圆锥折射这样的现象(更不用说激光了)。于是,汉密尔顿请他的实验学家朋友汉弗莱·劳埃德看看他能否找到这种新的折射——当他找到这种折射时,汉密尔顿一举成名,因为这可能是数学第一次被用来预测物理现象的存在,而不是解释已知的现象。随后出现了更多这样的突破:十年后,海王星的存在被预测出来,然后是一系列的预测,从无线电波和辐射压力到光子和E = mc 2到希格斯玻色子、引力波,等等。

Back in early 1832, however, no one had ever seen such a phenomenon as conical refraction (let alone lasers). So Hamilton asked his experimentalist friend Humphrey Lloyd to see if he could find this new kind of refraction—and when he did, Hamilton became a sensation, for it was perhaps the first time that mathematics had been used to predict the very existence of a physical phenomenon, rather than explaining what was already known. Many more such breakthroughs were to come: the prediction of the existence of Neptune would follow a decade later, and then a host of predictions, from radio waves and radiation pressure to photons and E = mc2 to the Higgs boson, gravitational waves, and much, much more.

• • •

• • •

汉密尔顿十几岁时就发表了第一篇关于光学的数学论文,而当他关于圆锥折射的数学预测得到验证时,他才 27 岁。而他做出这一惊人预测所需的工具之一就是微积分。

Hamilton had presented his first mathematical paper on optics when he was still a teenager, and he was still only twenty-seven when his mathematical prediction of conical refraction was verified. And among the tools he’d needed to make his remarkable prediction was calculus.

微积分有两个分支,微分和积分。微分学本质上是关于变化率的,比如速度,即距离随时间的变化率。在关于折射的研究中,汉密尔顿使用导数来显示某些量(与光程长度和波速有关)在晶体表面的变化,但所有形式的平滑变化现象可以用微积分来建模:加热和冷却、生物和原子的生长和衰变、各种波、生态系统、热和地形梯度、数学函数的梯度(也可用于解决优化问题)、金融趋势等等。

Calculus has two branches, differential and integral. Differential calculus is essentially about rates of change—as in speed, the rate of change of distance with time. In his work on refraction, Hamilton used derivatives to show how certain quantities (related to optical path length and wave speed) changed at the faces of the crystal, but all manner of smoothly changing phenomena can be modeled by differential calculus: heating and cooling, growth and decay both biological and atomic, waves of various kinds, ecological systems, thermal and topographical gradients, the gradients of mathematical functions (which can be used also to solve optimisation problems), financial trends, and so on.

至于积分,它是一种计算工具,用于解决古代数学家所面临的问题,例如计算田地和建筑平面图的面积或蓄水池和运河的体积。在当今的高科技社会中,还有更多这样的应用——例如计算飞机机翼升力面的面积;确定制造产品(从计算机和车身到桥梁和摩天大楼)所需材料的平方英尺数;确定建筑所用材料的重量(通过密度和体积),以确保结构具有足够的承重能力;在手术期间监测患者的血液量……我还可以继续列举下去,但您已经看到了范围。

As for integral calculus, it developed as a calculation tool to solve just the kinds of problems that ancient mathematicians had grappled with, when they wanted to calculate such things as the areas of fields and floor plans or the volumes of cisterns and canals. In today’s high-tech society there are many more such applications—calculating the area of the lifting surface of an airplane’s wing, for example; determining the square footage of material needed for manufacturing products, from computers and car bodies to bridges and skyscrapers; finding the weight (via the density and volume) of materials used in construction, to ensure the structure is sufficiently load-bearing; monitoring the volume of a patient’s blood during surgery … I could go on, but you can see the scope.

您可能最初是通过近似计算曲线y = f ( x ) 下的面积来学习积分的,其中f ( x ) 表示描述您感兴趣的曲线的x函数;例如,对于以原点为中心和基础的抛物线,则有f ( x )= x2。您可以使用许多细长的矩形来近似该面积,每个矩形的面积为ydx如图 2.1所示;添加的矩形越多,每个矩形就越细,近似值就越好。这是一种直观的几何方法,但是使用积分算法的代数公式,您可以“相加”无数个这些无穷小矩形来获得精确的面积。您可以做一些类似的事情来求体积,将无数个圆柱切片或壳相加,每个圆柱切片或壳的体积都是无穷小的。

You likely first learned integral calculus by approximating the area under a curve y = f(x), where f(x) represents the function of x that describes the curve you’re interested in; for instance, for a parabola centred and based at the origin, you’d have f(x) = x2. You’d approximate this area with a lot of skinny rectangles, each of area ydx, as in figure 2.1; the more rectangles you add, the thinner each one becomes and the better the approximation. It’s an intuitive geometrical approach, but with algebraic formulae for the algorithms of integral calculus, you can “add up” an infinite number of these infinitesimal rectangles to get the exact area. You do something similar to find volumes, adding up an infinite number of cylindrical slices or shells, each of infinitesimal volume.

“加一个无穷数……”这句话在今天听起来很容易,但能够操纵无穷数确实是一件了不起的事情。古代数学家完全回避了这个概念——就像他们对待我们之前遇到的虚数一样。

Saying “adding up an infinite number …” trips easily off the tongue today, but it’s really quite a marvel to be able to manipulate infinity. Ancient mathematicians had shied away from the concept altogether—just as they’d done with those imaginary numbers we met earlier.

例如,今天如果你想严格推导圆的面积(πr2)或周长(2πr)的公式,可以对其积分代数方程,如图 2.3所示。相比之下,三千五百年前,埃及抄写员阿赫梅斯在奇迹般保存至今的纸莎草卷轴上记录了一个用八边形近似圆形的简单几何结构。由此产生的面积计算结果令人惊讶地接近我们现在称为 π 的常数,如图2.2所示。要找到近似多边形的周长,阿赫梅斯只需将八边形的边加起来——使用毕达哥拉斯定理来找到角的长度。3

Today, for instance, if you wanted to derive rigorously the formula for the area (πr2) or circumference (2πr) of a circle, you’d integrate its algebraic equation, as we’ll see in figure 2.3. Three and a half thousand years ago, by contrast, the Egyptian scribe Ahmes recorded—on a papyrus scroll that miraculously still survives—a simple geometric construction that approximates a circle with an octagon. The resulting area computation gives a surprisingly good approximation for the constant we now call π, as you can see in figure 2.2. To find the circumference of the approximating polygon, Ahmes just had to add up the sides of the octagon—using Pythagoras’s theorem to find the lengths of the corners.3

图像

图 2.1。曲线下的面积。阴影矩形对应于图上的一般点 ( x, y ),其面积(长乘以宽)为ydx 。当你在x = 0 和x = b之间“积分”函数y = f ( x ) 时,当dx无穷小时,你实际上是在求所有这些矩形的总和。这个积分写为0bd,或者更具体地说,对于这个函数,0bfd,它等于当x值介于 0 和b之间时x轴与曲线f ( x )之间的面积。

FIGURE 2.1. Area under a curve. The shaded rectangle corresponds to the general point (x, y) on the graph, and its area (length by width) is ydx. When you “integrate” the function y = f(x) between x = 0 and x = b, you’re effectively finding the sum of all these rectangles when dx is infinitesimally small. This integral is written as 0bydx, or more specifically for this function, 0bfxdx, and it equals the area between the x-axis and the curve f(x) for x values between 0 and b.

在阿赫梅斯记录下他那个时代的数学一千年后,希腊数学家们发明了“穷举法”(后来被描述为穷举法),其中圆被多边形逐渐逼近,边数逐渐增加,直到达到所需的精度。这是严谨性方面的一次出色尝试——许多古代数学家对证明和计算的想法都很感兴趣——锡拉丘兹的阿基米德就是其中一位,他很好地利用了它。为了找到圆的周长,他从一个内接正六边形开始,然后将边数翻倍,比较两个连续多边形的周长。他不断地翻倍和比较,就这样,他找到了一种算法,用于添加和比较多达九十六条边的多边形的线段!这无疑是一项令人印象深刻的计算壮举,如果你想象自己试着画出它,你会发现九十六条边的多边形是一个漂亮的对圆的精确近似——阿基米德能够将(我们所说的)π 的值估计为介于3107131070上限是我们很多人在学校学到的价值观:317或 22/7(与现代值 3.14159 相比,为 3.14286,均四舍五入到小数点后五位)。

A thousand years after Ahmes recorded the maths of his day, Greekspeaking mathematicians developed “the method of exhaustion” (as it was later described), in which circles were successively approximated by polygons with an increasing number of sides, up to a desired accuracy. It was a brilliant attempt at rigour—many ancient mathematicians were interested in the idea of proof as well as in computation—and Archimedes of Syracuse, for one, made fine use of it. To find the circumference of a circle, he started with an inscribed regular hexagon, and then doubled the number of sides, comparing the perimeters of the two successive polygons. He kept on doubling and comparing, and, in this way, he found an algorithm for adding and comparing the segments of polygons with up to ninety-six sides! It was certainly an impressive feat of computation, and if you imagine trying to draw it, you’ll see that a ninety-six-sided polygon is a pretty fine approximation to a circle—good enough that Archimedes was able to estimate the value of (what we call) π as somewhere between 31071 and 31070. The upper limit is the value that many of us learned in school: 317 or 22/7 (which is 3.14286 compared with the modern value of 3.14159, both rounded to five decimal places).

不过,将九十六条边相加并不等同于将多边形变成真正的圆形所需的无数条边相加。相比之下,在微积分中,曲线的长度是通过对曲线的微小部分的长度代数公式进行积分来求得的,用ds表示。它是阿基米德九十六条边之一的无穷小版本,其长度是通过类比毕达哥拉斯定理得出的。对于图 2.3b中的示例,您可以看到,即使与 Ahmes 的方法也有祖先的相似之处。

Still, adding up ninety-six sides is not the same as adding up the infinite number of them required to turn the polygon into a true circle. With calculus, by contrast, lengths of curves are found by integrating the algebraic formula for the length of a minute segment of the curve, denoted by ds. It’s an infinitesimal version of one of Archimedes’s ninety-six sides, and its length is found by analogy with Pythagoras’s theorem. For the example in figure 2.3b, you can see there’s an ancestral similarity even with Ahmes’s approach.

图像

图 2.2。古埃及人对圆的面积和周长的估计。正方形的边长为 9 个单位,因此其面积为 81 平方单位。内接八边形的面积 = 正方形的面积减去四个角三角形的面积=8143×32=63平方单位。这似乎非常接近 64 平方单位,并且6463——因此,阿美斯将圆的面积记录为边长为 8 个单位的正方形的面积。由于八边形被视为原始正方形内接圆的近似值,因此圆的半径为92单位。所以他们的面积公式(用现代术语来说)是A = Nr 2,其中N代表他们的 π 版本:=一个r2=64922=64×481=25681,约为 3.16;现代值略高于 3.14。

FIGURE 2.2. Ancient Egyptian estimate of the area and circumference of a circle. The square has sides of length 9 units, so its area is 81 square units. The area of the inscribed octagon = area of square minus area of the four corner triangles =8143×32=63 square units. This seemed pretty close to 64 square units, and 64 was a lot easier to contemplate than 63—so Ahmes recorded the area of the circle as equivalent to the area of a square of side 8 units. Since the octagon is taken as an approximation of the circle inscribed in the original square, the circle has a radius of 92 units. So their area formula was (in modern terms) A = Nr2, where N stands for their version of our π: N=Ar2=64922=64×481=25681, which is about 3.16; the modern value is a little over 3.14.

将八边形的边相加(对角线使用毕达哥拉斯定理),并将其与半径为的圆的周长的现代公式进行比较92,你会得到埃及人对 π 的近似值为 3.2 — 仍然与他们的 3.16 大致相同。

Adding up the sides of the octagon (using Pythagoras’s theorem for the corners) and comparing this with the modern formula for the circumference of a circle of radius 92, you get the Egyptian approximation of π to be 3.2—still in the same ballpark as their 3.16.

图像

图 2.3A。使用积分计算圆的面积。其思路是求半圆下面的面积=r22并乘以 2。

FIGURE 2.3A. Finding the area of a circle using integral calculus. The idea is to find the area under the semicircle y=r2x2 and multiply by 2.

图像

图 2.3B。使用积分计算圆的周长。这里的思路是将所有小线段ds相加(积分!) 。(我将在相关方框中展示一些计算,但在这里和本书其他地方一样,如果您不喜欢这些细节,我希望您不要停止阅读,而是跳过它们并继续阅读故事。)

FIGURE 2.3B. Finding the circumference of a circle using integral calculus. Here the idea is to add up (integrate!) all the little segments ds. (I’ll show some of the calculations in the related box, but here as elsewhere in the book, if the details are not your thing, I hope you don’t stop reading but just skip over them and move on with the story.)

图 2.3A 的计算

CALCULATIONS FOR FIGURE 2.3A

选择极坐标,设x = r cos θ,因此对于固定半径r,dx = − r sin θ d θ;同样设y = r sin θ。(下一章的图 3.4显示了 sin θ 和 cos θ 的定义方式,尽管那里的半径为 1。)然后,使用双角公式

Choosing polar coordinates, let x = r cos θ, so for a fixed radius r, dx = −r sin θ dθ; also let y = r sin θ. (Fig. 3.4 in the next chapter shows how sin θ and cos θ are defined, although the radius there is 1.) Then, using the double angle formula

cos 2θ = (cos θ) 2 − (sin θ) 2 = 1 − 2(sin θ) 2 ,

cos 2θ = (cos θ)2 − (sin θ)2 = 1 − 2(sin θ)2,

并改变第二个积分中积分极限的顺序,使dx取负号,则半径为r的圆的面积为:

and changing the order of the limits of integration in the second integral to take in the minus sign in dx, the area of a circle of radius r is:

一个=2rrr22d=20πr正弦θrθdθ=2r20πθ2dθ=r20π1余弦2θdθ=r2θ2θ/20π=πr2

A=2rrr2x2dx=20πrsin θrsinθdθ=2r20πsinθ2dθ=r20π1cos2θdθ=r2θsin2θ/20π=πr2

您还可以使用二重积分(我们将在第 6 章中简要介绍)更轻松地做到这一点。

You can also do this, rather more readily, with a double integral (which we’ll meet briefly in chap. 6).

图 2.3B 的计算

CALCULATIONS FOR FIGURE 2.3B

再设x = r cosθ,y = r sin θ,则对于固定半径r,dx = r sin θ d θ 且dy = r cos θ d θ;则圆周的无穷小部分ds的长度为:

Again let x = r cosθ, and y = r sin θ, so that for a fixed radius r, dx = r sin θ dθ and dy = r cos θ dθ; then the length of ds, the infinitesimal segment of the circumference, is:

ds=d2+d2=r2余弦θdθ2+r2θdθ2=rdθ

ds=dx2+dy2=r2cosθdθ2+r2sinθdθ2=rdθ.

将此积分沿 θ = 0 到 θ = 2π 的整个圆进行积分,我们发现半径为r的圆的周长为 2π r

Integrating this around the whole circle from θ = 0 to θ = 2π, we find that the circumference of a circle of radius r is 2πr.

如果阿基米德没有在公元前 212 年罗马人围攻锡拉库萨时被杀,他可能更接近发明微积分。他当时大约 75 岁,杀死他的士兵显然不尊重长辈或数学——据说阿基米德沉浸在计算中,没有听到士兵的命令。两千年后,在另一场暴力动乱——法国大革命的黑暗日子里,一位了不起的年轻女孩读到了这个故事。她对阿基米德的迷恋感到非常兴奋,决定学习显然令人兴奋的数学学科——但她必须自学,因为当时大多数女孩都无法接受正规教育。而且她必须秘密学习,因为她的父母和几乎所有人一样,认为由于女性通常比男性体格更小、更弱,认真学习会损害她们的健康,阻止她们生育,甚至让她们发疯。她就是索菲·热尔曼,她成为二十世纪前最优秀的女数学家之一。她在最重要的工作中使用了微积分,开创了振动表面的数学理论。

Archimedes might have come even closer to inventing calculus if he hadn’t been killed during the Roman siege of Syracuse in 212 BCE. He was about seventy-five years old, and the soldier who killed him apparently had no respect for elders or for mathematics—Archimedes was reputedly so immersed in his calculations that he hadn’t heard the soldier’s orders. Two thousand years later, during the dark days of another violent upheaval, the French Revolution, a remarkable young girl read this story. She was so excited by Archimedes’s thrall that she decided to learn the evidently thrilling subject of mathematics—but she had to teach herself because most girls were denied a proper education at that time. And she had to do it in secret, because her parents believed, like almost everyone else, that because women were generally physically smaller and weaker than men, serious study would damage their health, stopping them from having babies or even driving them mad. She was Sophie Germain, and she became one of the best female mathematicians before the twentieth century. She used differential calculus in her most important work, pioneering the mathematical theory of vibrating surfaces.

尽管阿基米德的死很悲惨,但如果他最重要的手稿没有被中世纪的抄写员部分洗掉并改写,微积分也许会更早出现。直到 1906 年,手稿才被恢复,表明阿基米德确实发现了一些非凡的成果。仅凭具体几何和机械类比,他就设法找到了球体和曲面容器的体积,所用方法预示了如今本科积分微积分课上教授的“旋转体”的体积。但这项天才之作、这一信息宝藏却失传已久,以至于早期现代数学家不得不自己重新发现其中的思想。

Even despite Archimedes’s tragic death, calculus might have arrived sooner if his most important manuscript on the subject hadn’t been partly washed off and written over by a medieval scribe. It was recovered only in 1906, and it showed that Archimedes had found some truly extraordinary results. Working just with concrete geometry and mechanical analogies, he’d managed to find the volumes of spheres and curved-shaped containers, using methods that prefigure the volumes of “solids of revolution” taught today in undergrad integral calculus classes. But this work of genius, this treasure of information, was lost for so long that early modern mathematicians had had to rediscover its ideas for themselves.

牛顿、莱布尼茨和芝诺:寻找无穷小

NEWTON, LEIBNIZ, AND ZENO: SEARCHING FOR THE INFINITESIMAL

于是我们谈到了早期现代天才的化身——艾萨克·牛顿。根据牛津英语词典,“天才”的现代定义是“一种非凡的智力或创造力或其他自然能力或倾向”。这个词在日常用语中往往被过度使用,但它绝对适合牛顿,以及我们将在这个故事中遇到的其他极具创造力的数学家和数学物理学家——他们的思想如此创新,以至于在数学革命中发挥了奠基性作用,为我们带来了矢量和张量微积分。当然,即使是这些杰出的思想家也需要同事的启发和帮助,因为“孤独的天才”刻板印象很少适用于现实生活。然而,两百年来,牛顿被广泛奉为独一无二的天才——这种看法直到 19 世纪末才塑造和限制了人们对数学家和理论物理学家的认识。毕竟,我们不可能都是天才,历史上许多优秀的数学思想家都做出了重要贡献。我们将在故事中认识他们中的一些人,包括帮助这些天才取得突破的同事。

And so we come to that early modern incarnation of genius, Isaac Newton. According to the Oxford English Dictionary, the modern definition of “genius” is “an exceptional intellectual or creative power or other natural ability or tendency.” It’s a word that tends to be overused in everyday speech, but it certainly fits Newton, and the other exceptionally creative mathematicians and mathematical physicists we’ll meet in this story— the ones whose ideas were so innovative they played foundational roles in the mathematical revolutions that gave us vector and tensor calculus. Of course, even these standout thinkers needed inspiration and help from their colleagues, because the “lone genius” stereotype rarely applies in real life. For two hundred years, though, Newton was widely deified as singularly brilliant—a perception that shaped and limited the idea of what it meant to be a mathematician and a theoretical physicist until late in the nineteenth century. After all, we can’t all be geniuses, and many fine mathematical thinkers throughout history have made important contributions. We’ll meet some of them throughout this story, including the colleagues who helped the feted geniuses to make their breakthroughs.

牛顿天才的标志之一是,他对数学和物理学的许多分支做出了杰出的贡献:在现代从广义上讲,他是一位实验物理学家,尤其擅长光的研究;他是一位理论物理学家,尤其擅长运动和引力理论的研究;他是一位应用数学家(就他本人以及本文中的其他几位作者以及当今的许多研究人员而言,当数学应用于自然物理过程时,应用数学与理论物理学是相交叉的);他是一位纯数学家,他发明了新的数学方法、概念和证明,注重严谨性和美感。这些“纯”技术往往为应用数学家和理论物理学家后来的实际应用奠定了基础,微积分就是一个典型的例子。

One of the hallmarks of Newton’s genius is that he was an outstanding contributor to so many branches of mathematics and physics: in modern terms, he was an experimental physicist, especially on the subject of light; he was a theoretical physicist, especially with his theories of motion and gravity; he was an applied mathematician—a category that in his case, and for several others in this story as well as for many researchers today, intersects with theoretical physics when mathematics is applied to natural physical processes—and he was a pure mathematician, one who invents new mathematical methods, concepts, and proofs, with a focus on rigour and beauty. These “pure” techniques often underpin the later real-world applications of the applied mathematicians and theoretical physicists— and calculus is a classic example.

牛顿首次发现了微分和积分这两种微积分的通用符号表示算法,清楚地表明了它们之间的联系。也就是说,他证明了积分是微分的逆运算。戈特弗里德·莱布尼茨也意识到了这一点,因为微积分当时“尚在空中”,这要归功于哈里奥特和伽利略、皮埃尔·德·费马(费马最后定理的提出者)、艾萨克·巴罗(牛顿在剑桥的导师)和约翰·沃利斯等先行者。正是通过他们的工作以及他们所借鉴的工作,从阿基米德和欧几里得等古人到阿布·阿里·伊本·海赛姆和尼科尔·奥雷斯姆等中世纪学者,莱布尼茨和牛顿才得以各自独立地将微积分形成其第一个明确形式。

Newton was the first to find general, symbolically represented algorithms for the two types of calculus, differential and integral, clearly showing the connection between them. That is, he showed that integration is the inverse operation of differentiation. Gottfried Leibniz got the idea, too, for calculus was “in the air,” thanks to forerunners such as Harriot and Galileo, Pierre de Fermat (of Fermat’s last theorem fame), Isaac Barrow (Newton’s mentor at Cambridge), and John Wallis. It is through their works—and those they built on, from ancients such as Archimedes and Euclid to medieval scholars such as Abu Ali Ibn al-Haytham and Nicole Oresme—that Leibniz and Newton were able, each independently, to put calculus into its first definitive form.

这两位创始人截然不同。牛顿对他的发现讳莫如深,因为他无法忍受被批评。他对很多事情都很奇怪,尽管我确实同情他。他在微积分、光学引力理论方面取得了最初的突破,当时他只有二十出头——藏在母亲的农场里,在 1665-67 年的瘟疫年代与世隔绝。这样的成就很难不让人敬畏,也很难让人不为内心受惊的孩子感到难过。在他出生之前,他的父亲就在奥利弗·克伦威尔的议会党和查理一世的保皇党之间的内战中去世了。对于他的母亲汉娜来说,那一定是一段可怕的时光,她的丈夫死了,战斗如此激烈——距离她的农场只有附近的村庄。于是她嫁给了富裕的牧师巴纳巴斯·史密斯,并搬到了另一个城镇;不幸的是——显然是在史密斯的坚持下——她把年幼的艾萨克和她的母亲留在了那里。众所周知,他希望自己能把史密斯的房子烧毁,把史密斯和他母亲都烧了,但他似乎唯一表面上的破坏性行为就是偶尔画一些艺术涂鸦,或者偶尔在晚上放爆炸的风筝恶作剧,把迷信的村民吓得魂飞胆丧。直到史密斯去世后,汉娜才带着三个不受欢迎的同父异母兄弟姐妹回到艾萨克身边,当时艾萨克大约十岁。4

These two founders could not have been more different. Newton was rather secretive about his discoveries because he couldn’t stand the thought of being criticised. He was rather weird about a lot of things, although I do have some sympathy for him. He made his initial breakthroughs with calculus, optics, and the theory of gravity when he was still only in his early twenties—tucked away at his mother’s farm, isolating during the plague years 1665–67. It’s hard not to be awed by such an achievement. And it’s hard not to feel for the frightened child inside. Before he was born, his father had died in the Civil War between Oliver Cromwell’s Parliamentarians and King Charles I’s Royalists. It must have been a terrifying time for his mother, Hannah, with her husband dead and the fighting so fierce—it came as close to her farm as the nearby village. And so she married Barnabas Smith, a well-off rector, and moved to another town; unfortunately—and apparently at Smith’s insistence—she left young Isaac behind with her mother. He famously wished he could have burned Smith’s house down with his mother and Smith inside, but it seems the only outwardly destructive behavior he engaged in was the odd bit of artistic graffiti or the occasional night-time pranks with exploding kites that frightened the wits out of superstitious villagers. Only when Smith died did Hannah return to Isaac, then aged about ten, with three unwelcome half-siblings in tow.4

莱布尼茨比牛顿阳光得多。莱布尼茨学者菲利普·维纳说,莱布尼茨是一位真正的文艺复兴人,他“是一位律师、科学家、发明家、外交官、诗人、语言学家、逻辑学家,也是一位虔诚地捍卫理性培养的哲学家,认为理性培养是人类进步的光辉希望” 。5

Leibniz was a much sunnier person than Newton. A true Renaissance man, he was “a lawyer, scientist, inventor, diplomat, poet, philologist, logician, and a philosopher who religiously defended the cultivation of reason as the radiant hope of human progress,” to quote Leibniz scholar Philip Wiener.5

莱布尼茨六岁时,身为哲学教授的父亲去世。莱布尼茨的外祖父也曾是法学教授,年轻的戈特弗里德后来不仅获得了法学学位,还获得了博士学位。相比之下,牛顿的父亲是一位目不识丁的农民,他的母亲并不理解他对知识的热爱和天赋:她在他十七岁时就让他辍学,直到他把农场经营得一团糟,母亲最终接受了校长和她在剑桥大学毕业的哥哥的建议,允许艾萨克为上大学做准备。但他必须自费完成剑桥学业,为一位富有的学生做“sizar”或仆人。对他和我们来说幸运的是,他很快就成为了那里的教授,可以自由地做出他惊人的发现。

When Leibniz was six, his father, a philosophy professor, died. His mother’s father had also been a professor—of law, in his case, the field in which young Gottfried later earned not just a degree but also a doctorate. By contrast, Newton’s father had been an illiterate farmer, and his mother had not understood his intellectual passion and talent: she took him out of school when he was seventeen, until he made such a mess of running the farm that she finally accepted the advice of both the schoolmaster and her Cambridge-educated brother, and allowed Isaac to prepare for university. But he had to pay his own way through Cambridge, working for a wealthy student as a “sizar” or servant. Fortunately for him and for us, he soon became a professor there, and was free to make his amazing discoveries.

莱布尼茨在学生时代没有这样的烦恼,毕业后他过着相当光鲜的生活。他为他的王室赞助人在欧洲执行外交任务,一路上与科学家和哲学家交谈,建立了一个由大约六百名通信者组成的庞大网络——他是一位多产的书信作家,对世界和平、哲学和科学有着远大的想法。鉴于他广泛的兴趣和职业职责,他表现出非凡的数学天赋。毫不奇怪,学术上的牛顿是更优秀的数学家和实验物理学家,但兼收并蓄的莱布尼茨仍然做出了重大发现——他是更好的符号制造者。

Leibniz had no such worries as a student, and after graduating he led a rather glamorous life. Traveling through Europe on diplomatic missions for his princely patrons and conversing with scientists and philosophers along the way, he built up a huge network of some six hundred correspondents—he was a prolific letter writer, with big ideas about universal peace, philosophy, and science. Given all his wide-ranging interests and his professional duties, he showed an extraordinary talent for mathematics. Not surprisingly, the academic Newton was the better mathematician and experimental physicist, but the eclectic Leibniz made significant discoveries nonetheless—and he was the better symbol-maker.

在上一章中,我强调了代数从文字到符号的缓慢发展,但在现代微积分的早期——在维埃特、哈里奥特和笛卡尔引入真正的代数符号主义半个世纪之后——情况发生了逆转:微积分符号代表了当时没有人能够用文字充分解释的想法。这就是符号思维的特殊力量:它可以带你进入最初超出普通理解的新领域。

In the previous chapter, I highlighted algebra’s slow progress from words to symbols, but in the early days of modern calculus—half a century after Viète, Harriot, and Descartes introduced true algebraic symbolism— the situation was reversed: calculus symbols stood for ideas that no one at the time could adequately explain in words. This is the special power of symbolic thinking: it can take you into new places that are, at first, beyond ordinary understanding.

例如,尽管我一直在轻松地谈论“无穷大”和“无穷小”,但它们是很难有意义地定义的概念。现代词典对“无穷小”的定义是“极其小”,但这对数学家没有多大帮助。他们需要更精确的定义,尽管在这种情况下,在牛顿-莱布尼茨微积分诞生两百年后才找到这样的定义(通过极限、连续性和函数理论)。当你考虑到要找到一个运动物体在任何给定时间的速度,你必须将它当时的位置与一瞬间之后的位置进行比较,然后将差值除以时间的瞬间时,你就会明白其中的困难。但你所说的“瞬间”是什么意思呢?你如何定义位置的增量变化?显然,它们都是非常小的量,但有多小呢?

For instance, although I’ve been talking easily about “infinity” and “infinitesimal” quantities, they are concepts that are very difficult to define meaningfully. A modern dictionary definition of “infinitesimal” is “extremely small,” but that’s not much help to mathematicians. They need something more precise, although in this case it took two hundred years after the birth of Newtonian-Leibnizian calculus to find such a definition (via the theory of limits, continuity, and functions). You can see the difficulty when you consider that to find a moving object’s speed at any given time, you have to compare its position at that time with its position an instant later, and then divide the difference by the instant of time. But what do you mean by an “instant”? And how do you define the incremental change in position? Obviously, they are both very small quantities, but how small?

牛顿将这些无穷小的变化称为“矩”,用符号 ο 表示(希腊字母 omicron 表示“不完全为零”)。莱布尼茨将它们称为“差”或“微分”,并将它们表示为dt、dx等。对于距离x相对于时间的变化率,牛顿使用一个点,如。另一方面,莱布尼茨则用“比率”来表示,例如x相对于t的变化率,以及y相对于作为dddd等等。实际上,莱布尼茨主要使用dy : dx符号,其中冒号表示比率;是他的追随者——尤其是约翰·伯努利(他的名字通常以法语方式写成 Jean,有时也写成 John)——将这个比率推广为dd。你可以看到谁的符号最终胜出——尽管点符号仍然用于区分时间,这对于研究运动、波、场以及随时间变化的其他各种量非常重要。

Newton called these infinitesimal changes “moments,” denoting them by the symbol ο (a “not quite zero” represented by the Greek letter omicron). Leibniz called them “differences” or “differentials,” and denoted them dt, dx, and so on. For the rate of change of a distance x with respect to time, Newton used a dot, as in . Leibniz, on the other hand, wrote his “ratios”—his rates of change of x with respect to t, for instance, and of y with respect to xasdxdt,dydx, and so on. Actually, Leibniz mostly used the notation dy:dx, where the colon symbolised a ratio; it was his followers—especially Johann Bernoulli, whose first name is commonly written in the French way, Jean, and occasionally Anglicised as John—who popularised this ratio as dydx. You can see whose symbols eventually won the day—although the dot notation is still used for differentiation with respect to time, which is important in the study of motion, waves, fields, and various other quantities that change over time.

牛顿和莱布尼茨都意识到了“无穷小”一词背后的概念问题。莱布尼茨尝试通过改进古老的穷举法来解释它,而牛顿则对我们现在所说的极限做出了合理的早期定义。(您可以在尾注6中看到他们的尝试,以及与之相对的现代定义,我在下面的图 2.4中勾勒出了极限的概念。)然而,在计算中,莱布尼茨和牛顿经常做任何精明的学生都会做的事情:忽略这些微小的增量和瞬间,因为它们太小而不必担心——有点像你花 124.99 美元买一件商品,不用担心 125.00 美元的零钱。它在实践中很有效,只是像一分钱一样,“瞬间”的时间实际上并不为零。如果你把它当成,当你想除以“瞬间”时,就会遇到理论问题:例如,要找到瞬时速度,你就得做不可能的事,除以零。更糟糕的是,距离的微小变化也近似为零,所以你会试图计算 0/0。

Both Newton and Leibniz were aware of the conceptual problem underlying the word “infinitesimal.” Leibniz had a stab at explaining it by refining the ancient method of exhaustion, and Newton made a reasonable fist of an early definition of what we now call limits. (You can see their attempts, and by comparison a modern definition, in the endnote,6 and I’ve sketched the limit idea in fig. 2.4 below.) In their calculations, though, Leibniz and Newton often did what any canny student would do: ignore these tiny increments and instants as too small to worry about—a little like when you buy an item for $124.99 and don’t worry about the change out of $125.00. It works well in practice, except that like a cent, an “instant” of time is not actually zero. If you act as though it is zero, you run into theoretical problems when you want to divide by an “instant”: to find an instantaneous speed, for example, you’d be doing the impossible and dividing by zero. Worse, the tiny change in distance is approximately zero, too, so you’d be trying to calculate 0/0.

阿基米德发现,将无穷小量相加已经够难了,更不用说将它们相除求导数了——早在公元前 450 年,传奇人物芝诺就在其著名悖论中强调了这一困难。例如,他说,要从 A 跑到 B,首先你必须跑到 B 的一半路程;但在跑到一半之前,你必须跑四分之一,以此类推,直到无限。这意味着你永远无法真正开始。当然,每个人都知道你可以从 A 跑到 B。芝诺似乎在这里表明——至少从现代的角度来看——运动的概念是多么难以定义,因为它的距离是无穷小的,时间是微小的“瞬间”。当然,芝诺本人可能有神秘的目的,而不是数学的目的,所以他的悖论——就像他同时代的佛教徒的教义一样——可能是为了表明时间和空间是幻觉。

It was hard enough trying to add infinitesimal quantities, as Archimedes found, let alone divide them to find derivatives—and as long ago as 450 BCE, the legendary Zeno had highlighted the difficulty in his famous paradoxes. For instance, he said, to run from A to B, first you have to run halfway to B; but before you reach halfway, you have to run a quarter of the way, and so on, indefinitely. Which means you can never really get started. Of course, everyone knows that you can run from A to B. What Zeno seemed to be showing here—from a modern perspective, at least—was just how difficult it is to define the idea of motion, with its infinitesimal increments in distance and its tiny “instants” of time. Of course, Zeno himself may have had a mystical rather than a mathematical purpose, so it’s possible that his paradoxes—like the teachings of his Buddhist contemporaries— were intended to suggest that time and space are illusions.

航行公海并破解战时密码:代数微积分的兴起

SAILING THE HIGH SEAS AND BREAKING WARTIME CODES: THE RISE OF ALGEBRAIC CALCULUS

数学家们花了近两千五百年的时间才理清极限理论,以便对下列问题给出令人满意的解答:芝诺悖论以及微积分。第一步涉及“收敛无穷级数”的概念,其中确实有可能将无限数量的越来越小的项相加,并从数学上证明结果是有限的。数学爱好者可能会认识到芝诺的 A 到 B 问题是几何级数12+14+18+...,等于 1——所以你确实可以穿越 A 和 B 之间的整个距离!不过,并非所有的无穷级数都会“收敛”(或加起来等于一个有限数),因此极限理论对于区分二者至关重要。积分微积分也需要将无限数量的量相加,因此你可以看到它的历史与无穷级数的早期研究密切相关。

It took mathematicians almost two and a half thousand years to sort out the theory of limits, which is needed to give a satisfactory resolution of Zeno’s paradoxes as well as of calculus. One of the first steps involved the idea of “converging infinite series,” where it does indeed turn out to be possible to add up an infinite number of increasingly small terms and to mathematically prove that the result is finite. Maths buffs might recognise that Zeno’s A to B problem is the geometric series 12+14+18+...,, which equals 1—so you can, indeed, traverse the whole distance between A and B! Not all infinite series “converge” (or add up to a finite number), though, so limit theory is essential in order to tell the difference. Integral calculus, too, requires adding up an infinite number of quantities, so you can see that its history is intimately linked to the early work on infinite series.

尽管极限和收敛的概念直到现代才被严格证明,但古代和中世纪的数学家已经使用巧妙的几何结构来求有限个级数的和。他们用文字写下级数和算法——哈里奥特似乎是第一个像我们今天一样用数字和字母符号重写级数的人,他在 1600 年左右做到了这一点。特别是在一次漫长而精湛的计算中,他使用无限几何级数和直观的极限论证来找到一艘船沿着固定罗盘方位在弯曲的地球上绘制的螺旋线长度的代数表达式。(他的赞助人沃尔特·雷利爵士希望获得最好的科学专业知识来帮助提高航行的安全性和效率,因为他正在计划从英国到美国的未知海洋的航行。)哈里奥特首先将螺旋线分解成小段,就像阿基米德的圆一样,但他假设它们的数量是无限的,并且大小不断减小。他花了数十页纸和数月的时间才完成这项工作——这是除早期对圆周长的估计之外,第一个现存的精确曲线长度推导。如今,积分微积分用于求出这种长度,如图2.3b所示。不过,哈里奥特会很高兴知道他的代数符号方法直接影响了英国微积分先驱约翰·沃利斯,而沃利斯又影响了年轻的牛顿。7

Although the ideas of limits and convergence weren’t proven rigorously until modern times, ancient and medieval mathematicians had used ingenious geometric constructions to find the sums of a limited number of series. They wrote both their series and their algorithms in words— Harriot seems to have been the first to rewrite series in terms of number and letter symbols as we do today, which he did sometime around 1600. In particular, in a long and virtuoso calculation, he used an infinite geometric series and an intuitive limit argument to find an algebraic expression for the length of the spiral traced over the curved Earth by a ship following a fixed compass bearing. (His patron, Sir Walter Ralegh, had wanted the best scientific expertise to help make navigation safer and more efficient, for he was planning voyages from England to America across an uncharted ocean.) Harriot began by breaking up the spiral into small segments like Archimedes’s circle, except he assumed an infinite number of them, of ever-decreasing size. He took dozens of pages and many months to conclude this work—the first extant exact derivation of the length of a curved line, other than those early estimates of the circumference of a circle. Today, integral calculus is used to find such lengths, as figure 2.3b showed. Still, Harriot would have been chuffed to know that his symbolic approach to algebra directly influenced the English calculus pioneer John Wallis, and Wallis influenced the young Newton.7

图 2.3b和相关方框还表明,要使用微积分来计算曲线的长度,你需要一个曲线公式——这就是解析几何的领域。哈里奥特也开创了这一领域,但真正将“解析”几何或代数几何发扬光大的人是笛卡尔,为了纪念他,我们将矩形x、y、z坐标系称为“笛卡尔坐标系”。即便如此,笛卡尔也几乎没能意识到用以这些坐标写出的代数方程来表示几何曲线的非凡威力(例如上图 2.3中圆的代数方程为x 2 + y 2 = r 2)。他的主要目的是表明代数和几何结合起来可以使解决问题变得更容易:代数符号使大脑不必将复杂的几何构造形象化(例如卡尔达诺的完整立方体),而几何则为代数运算赋予具体的含义。二十年后,沃利斯对此事不那么公正,他绝对更喜欢代数而不是几何。

Figure 2.3b and the related box also showed that to use calculus to find the length of a curved line, you need a formula for your curve—and this is the province of “analytic geometry.” Harriot had made a start at this, too, but it was Descartes who really put “analytic” or algebraic geometry on the map, and we call the rectangular x, y, z coordinate system “Cartesian” in his honour. Even so, Descartes barely glimpsed the remarkable power of being able to represent geometric curves by algebraic equations written in terms of these coordinates (such as x2 + y2 = r2 for the circle in fig. 2.3 above). His main purpose had been to show that algebra and geometry together make problem-solving easier: algebraic symbolism frees the mind from having to visualise complicated geometric constructions (such as Cardano’s completed cube), while geometry gives concrete meaning to algebraic operations. Twenty years later, Wallis was less evenhanded about the matter, definitely favouring algebra over geometry.

沃利斯之所以更喜欢英国人而不是法国人,是因为他对于法国人(以费马和其他几个人为代表)对他早期著作的反应感到恼火。他还发誓,而且他并不是唯一一个发誓笛卡尔抄袭了哈里奥特的《实践》一书的人。8无论如何,沃利斯在 1655 年出版的《无穷算术》一书中向代数微积分迈出了重要一步。他甚至创造了一个表示无穷的符号——就是我们今天使用的 ∞。罗马人曾偶尔使用这个符号来表示 1,000,它也时不时地出现在各种语境中,包括埃及图像的一个版本,图中一条蛇在吃着自己的尾巴,象征着生死无尽的循环。没有人知道沃利斯为什么用它来表示无穷;也许他对它的古老神秘含义印象深刻,尽管作为加尔文教徒,他选择它更有可能只是因为它代表了一条永无止境的曲线。

Wallis also favoured the English over the French, because he’d been miffed by the French response—in the persons of Fermat and several others—to his own early work. He also swore, and he wasn’t entirely alone in this, that Descartes had cribbed from Harriot’s book Praxis.8 Anyway, Wallis took a significant step toward an algebraic calculus in his 1655 book Arithmetica Infinitorum. He’d even created a symbol for infinity—the same one we use today, ∞. This symbol had been used occasionally by the Romans to represent 1,000, and it also cropped up every now and then in various contexts, including a version of the Egyptian image of a snake eating its own tail and symbolising the endless cycle of birth and death. No one knows why Wallis used it for infinity; perhaps he was impressed with its ancient mystical connotations, although as a Calvinist he more likely chose it simply because it represents a never-ending curve.

沃利斯的故事让我们得以一窥当时的数学和政治背景,就在牛顿崭露头角之前。他是肯特郡一位乡村牧师的儿子,幸运的是,他的家人认可并支持他的学术倾向。他最终去了剑桥,尽管在那里他没学到多少数学。直到圣诞节假期回家时,他才意识到这门学科的存在,当时他注意到弟弟正在学习算术,为做职业做准备。他很感兴趣,要求上一堂课。正如他后来所说,

Wallis’s story gives a glimpse into the mathematical and political context at that time, just before Newton burst onto the scene. He was the son of a village rector in Kent, and fortunately his family recognised and supported his academic bent. He eventually went to Cambridge, although he didn’t learn much maths there. He’d only realised the subject existed when he came home for the Christmas break and noticed his younger brother studying arithmetic in preparation for a trade. Intrigued, he’d asked for a lesson. As he put it later,

那时,数学并不被视为学术知识,而是商人、海员、木匠、土地测量员或类似人员的工作……当时我们学院(剑桥大学伊曼纽尔分校)有 200 多名学生,我不知道有哪两个人比我更懂数学,而我懂的数学知识非常少;直到我被任命为数学教授之前,我从未认真学习过数学(除了作为一种愉快的消遣)。9

Mathematicks were not, at that time, looked upon as Academical Learning, but the business of Traders, Merchants, Sea-men, Carpenters, land-measurers, or the like.... Of more than 200 at that time in our College [Emmanuel, Cambridge], I do not know of any two that had more Mathematicks than myself, which was but very little; having never made it my serious studie (otherwise than as a pleasant diversion) till some little time before I was designed for a Professor in it.9

人们常说,学习某种东西的最好方法就是教它!而沃利斯肯定学得很好,因为他成为了牛顿之前那一代最优秀的数学家之一。然而,这真是一段不平凡的旅程。首先,他获得了神学硕士学位。然后,他获得了剑桥大学皇后学院的研究员职位,但是这份职位并没有维持多久:因为当时研究员必须保持未婚状态,因此在 1645 年,新婚的沃利斯成为了一名牧师而不是学者。然后,出乎意料的是,他发现自己有破译密码的天赋。当时南北战争正在激烈进行,一位牧师同事向他展示了一条被截获的密码信息,半开玩笑地问他能否理解。令他自己惊讶的是,沃利斯在几个小时内就破译了这条信息。这是他人生的转折点,因为当国王于 1649 年被处决时,如果你能证明自己对胜利者的忠诚,那将非常有利。沃利斯为议会党破解了许多密码,还展示了他的数学技能——当牛津大学的萨维尔几何学教授因保皇党而被解雇时,沃利斯被任命。(尽管沃利斯支持议会党,但他曾公开反对处决国王,并相当勇敢地签署了一份抗议文件。因此,当君主制后来恢复时,他仍然支持。此外,他更喜欢数学而不是政治,而且正如他告诉保皇党朋友的那样,他解开的大多数信息对议会党来说根本没多大用处。)10

They do say that the best way to learn something is to teach it! And Wallis must have learned it well, for he became one of the best mathematicians in the generation before Newton. It had been quite a journey, though. First, he graduated with a master of arts in divinity. Then he took up a fellowship at Queen’s College, Cambridge, but it was short-lived: fellows had to remain unmarried in those days, so in 1645, the newly wedded Wallis became a minister instead of an academic. And then, quite out of the blue, he discovered he had a knack for code breaking. The Civil War was raging, and a fellow clergyman had shown him an intercepted message in cipher, asking half-jokingly if he could make sense of it. To his own amazement, Wallis was able to decipher the message within a few hours. It was a turning point in his life, for when the king was executed in 1649, it was very handy if you’d proved your loyalty to the victors. Wallis had broken a number of codes for the Parliamentarians, showing his mathematical skills into the bargain—and when Oxford’s Savilian Professor of Geometry was sacked for having been a Royalist, Wallis was appointed instead. (Although he supported the Parliamentarians, Wallis had spoken out against the execution of the king, and had signed, rather bravely, a document protesting it. So when the monarchy was later restored, he remained in favour. Besides, he preferred mathematics to politics, and as he told his Royalist friends, most of the messages he’d unlocked hadn’t been of all that much use to the Parliamentarians anyway.)10

沃利斯在牛津大学任职仅仅六年后就发表了他的代数著作《无限算术》。当时,大多数数学家们仍然喜欢使用有形的几何思想来开发处理无穷和和无穷小增量的方法——托马斯·霍布斯在这方面尤其直言不讳。他今天的名声来自于他的政治哲学——以及他得出的严肃结论:没有国家的保护,生活是“肮脏、野蛮和短暂的”。但他也涉足数学,他表达了这种支持几何学的情绪,称沃利斯的《无限算术》是一本“肮脏的书”和“一堆符号”。事实上,他谴责“所有将代数应用于几何学的人”。显然,他不是一个开朗的人。

It was just six years after his appointment at Oxford when Wallis published his algebraic Arithmetica Infinitorum. At the time, most mathematicians still preferred using tangible geometric ideas to develop ways of handling infinite sums and infinitesimal increments—and Thomas Hobbes was particularly vociferous on the subject. His fame today rests on his political philosophy—and his dour conclusion that life is “nasty, brutish, and short” without the protection of a state. But he also dabbled in mathematics, and he expressed this pro-geometrical sentiment by pronouncing Wallis’s Arithmetica Infinitorum a “scurvy book” and “a scab of symbols.” In fact, he denounced “the whole herd of them who apply their algebra to geometry.” He evidently was not a sunny person.

尽管霍布斯和他的几何学家同行们对此持反对态度,但十年后,沃利斯的著作让年轻的牛顿首次踏上了学习微积分的道路。二十年后,牛顿写出了他那本宏伟的《自然哲学的数学原理》,英文全称是《自然哲学的数学原理》;和当时大多数欧洲学者一样,牛顿用拉丁文写作。我要补充一点,“物理学家”一词是十九世纪的;在此之前,理论物理学被称为“自然哲学”。事后看来,两者的区别在于,自然哲学注重逻辑论证,而理论物理学提供可以通过实验检验的解释和预测——这是牛顿率先改变的观点。例如,传统的自然哲学家援引了“逻辑”假设,即某种机械的东西,如巨大的空灵漩涡,将行星带到它们的轨道上。除了它们声称要“解释”的行星运动之外,没有任何实际证据表明这些漩涡存在。相比之下,牛顿根据可观察的事实对行星运动做出了解释,并由此得出了任何两个物体(包括行星和太阳)之间引力的可测试公式:引力与它们质量的乘积成正比,与它们之间的距离成反比。当然,用符号代数形式来表达,这就是著名的平方反比定律F=r2,其中质量为mM,假设距离为rG为比例常数。牛顿也是在《自然哲学的数学原理》中首次发表了他的通用微积分算法,并将其用于他的运动科学中。这些运动定律,加上引力定律,是如此普遍,以至于它们解释了一切从一片树叶的飘落到潮汐的引力,从空中抛出的球的轨迹到行星的运动。这是一项惊人的成就。

Despite Hobbes and his fellow geometers, a decade later it was Wallis’s book that first set young Newton on his path to calculus. Two decades after that, Newton wrote his magnificent Principia, whose full title, in English, is Mathematical Principles of Natural Philosophy; like most European scholars at the time, Newton wrote in Latin. I should add here that the term “physicist” is a nineteenth-century one; before then, theoretical physics was called “natural philosophy.” The difference, in hindsight, is that natural philosophy focussed on logical arguments, while theoretical physics offers explanations and predictions that can be experimentally tested—a change of viewpoint that Newton spearheaded. For example, traditional natural philosophers invoked the “logical” hypothesis that something mechanical, such as gargantuan ethereal whirlpools, carried the planets in their orbits. There was no actual evidence these whirlpools existed, aside from the planetary motion they purported to “explain.” By contrast, Newton had worked out an explanation for planetary motion based on observable facts, from which he derived a testable formula for the gravitational attraction between any two bodies, including a planet and the Sun: it is proportional to the product of their masses and inversely proportional to the distance between them. In symbolic algebraic form this is, of course, the famous inverse square law F=GmMr2, where the masses are m and M, say, the distance is r, and G is the constant of proportionality. It was also in Principia that Newton first published his general calculus algorithms, which he used in his science of motion. These laws of motion, together with the law of gravity, were so general that they explained everything from the fall of a leaf to the pull of the tides, from the path of a ball thrown in the air to the motion of the planets. It was an astonishing achievement.

爱因斯坦详细解释了这一成就的重要性:“在牛顿之前,”他说,“不存在关于经验世界深层特征的自足的物理因果关系系统。”例如,他补充道,开普勒基于观察的行星运动定律,包括行星轨迹的椭圆形状,确定了行星如何运动,但没有确定为什么运动。这是因为“这些定律关注的是整个运动,而不是系统的运动状态如何引起紧随其后的运动的问题;我们现在应该说,它们是积分定律,而不是微分定律。” 11我们很快就会看到牛顿的微分第二运动定律,稍后我们还会看到,这种对微分定律而非积分定律的偏好是詹姆斯·克拉克·麦克斯韦成功揭开光的秘密的关键。

Einstein explained in detail the importance of this achievement: “Before Newton,” he said, “there existed no self-contained system of physical causality [about] the deeper features of the empirical world.” For instance, he added, Kepler’s observation-based laws of planetary motion, including the elliptical shape of planetary trajectories, established how the planets move, but not why. That’s because “these laws are concerned with the movement as a whole, and not with the question how the state of motion of a system gives rise to that which immediately follows it in time; they are, as we should say now, integral and not differential laws.”11 We’ll see Newton’s differential second law of motion shortly, and later we’ll see that this same preference for differential over integral laws is key to James Clerk Maxwell’s success in uncovering the secret of light.

枯燥乏味的工作和剽窃: 《原理》的接受

DRY DRUDGES AND PLAGIARISM: THE RECEPTION OF PRINCIPIA

当时,获得如此热烈的认可并非易事。《数学原理》于 1687 年出版后,立即招致了批评,其中就包括莱布尼茨。和大多数同时代人一样,莱布尼茨也为牛顿的数学天才所折服,但作为一名哲学家,莱布尼茨对理论应做什么的看法与牛顿(以及后来的爱因斯坦)不同。牛顿曾解释说,行星运动的原因是引力,但莱布尼茨认为引力理论也应该解释引力本身。特别是,莱布尼茨认为引力理论应该提出一种具体的机制,比如那些空灵的漩涡,来解释引力如何物理地影响牛顿用数学描述的行星运动的瞬时变化。由于自然哲学家长期以来一直以这种方式定义科学理论的标志,许多其他批评家也同意莱布尼茨的观点。但牛顿开创了现代观点:引力显然存在,他量化了引力的影响——尤其是对行星运动的影响——但他不想加入无法验证的关于引力的猜测他当然想解释引力本身,但他知道他取得的成就在概念和范围上都是前所未有的,而且需要后来的研究人员对引力的本质有更深入的理解。(这就是爱因斯坦和他对张量的开创性运用进入我们故事的地方。)

Such glowing recognition didn’t come easily at the time. As soon as Principia was published in 1687 it had its detractors—including Leibniz. Like most of his contemporaries he was dazzled by Newton’s mathematical genius, but as a philosopher first and foremost, he had a different idea from Newton (and later Einstein) about what a theory should do. Newton had explained that the cause of planetary motion was gravity, but Leibniz thought that a theory of gravity should also explain gravity itself. In particular, he thought it should suggest a concrete mechanism—such as those ethereal whirlpools—for how gravity physically effected the momentby-moment changes in planetary motion that Newton had described mathematically. Since this is how natural philosophers had long defined the hallmark of scientific theory, many other critics agreed with Leibniz. But Newton inaugurated the modern view: it was obvious that gravity exists, and he’d quantified its effects—notably on planetary motion—but he didn’t want to include untestable speculation about what it was. He certainly would have liked to explain gravity itself, but he knew that what he had achieved was unprecedented in its conception and scope, and that it would be up to later researchers to find a deeper understanding of the nature of gravity. (That’s where Einstein, and his groundbreaking use of tensors, will come into our story.)

比这些批评更让牛顿感到苦恼的是罗伯特·胡克 (Robert Hooke) 指控他抄袭。胡克在科学家中名声不佳,因为他喜欢自吹自擂——而且他似乎特别喜欢公开批评牛顿。不过,最近,历史学家对他的作品进行了更仔细的研究,似乎他应该因对行星运动的引力解释提出一些重要的原创性想法而受到赞扬——包括所有行星都有引力,并且引力“可能”与距离成平方反比减弱,因为光的强度就是以这种方式减弱的。尽管牛顿之前已经思考了这个问题很多年,但正是胡克在 1679 年写给他的信激发了他对这个问题的思考——正如他在《数学原理》第一版中承认的那样。胡克提出了轨道运动由两种不同的“运动”组成的想法,一种是切向轨道的运动,另一种是指向运动中心的运动。这是一个直观的想法,即运动是由类似矢量的分量构成的,尽管它并不完全正确:牛顿会表明是向内的,但运动(速度)是切向的。

Even more distressing for Newton than these criticisms was Robert Hooke’s accusation of plagiarism. Hooke hasn’t had a good rap among scientists, because he tended to big-note himself—and he seemed to take special delight in criticising Newton publicly. Recently, though, historians have taken a closer look at his work, and it seems he deserves credit for some significant original thinking about a gravitational explanation of planetary motion—including the idea that all planets have gravity, and that it weakens with distance “probably” in inverse square proportion, because that is the way light’s intensity weakens. Although Newton had been thinking about the problem for years beforehand, it was Hooke’s letters to him, in 1679, that catalysed his thinking on the subject—as he acknowledged in the first edition of Principia. Hooke suggested the idea that orbital motion was made up of two different “motions,” one tangential to the orbit and one directed to the centre of the motion. It was an intuitive idea of motion being made of vector-like components, although it wasn’t quite correct: Newton would show that the force is directed inward but the motion (velocity) is tangential.

胡克没有将引力扩展到行星以外的物体,他无法通过直接实验或计算证明平方反比定律。相反,他根据对机械模型运动的仔细观察,做出了一些巧妙的类比。12牛顿不同,他没有将行星运动推广到所有类型运动的数学定律——因此,他没有像牛顿那样推导出我们现在所说的动能和势能的守恒。(莱布尼茨也开创了能量守恒的概念。)胡克也没有用他的引力概念来估计地球的扁圆形,也没有解释潮汐、春分点进动或牛顿解释的任何其他奥秘。因此,当胡克指责他抄袭时,牛顿非常愤怒:虽然胡克的数学能力比人们长期以来认为的要强,但根本无法与牛顿相提并论。难怪牛顿勃然大怒埃德蒙·哈雷说,如果胡克可以自称是引力理论的发明者,那么“发现、解决并做所有事情的数学家们就只能满足于做枯燥的计算者和苦力”。他说得有道理。有趣的是,在今天的版权法中,受保护的是表达形式而不是思想。尽管最近数学教育者中出现了一个重要举措,强调计算在数学历史和应用中的作用,但许多现代数学家与牛顿保持一致,在他们的工作中看到了美和创造力,而不仅仅是计算和应用。13

Hooke didn’t extend gravity to material bodies other than planets, and he wasn’t able to prove the inverse-square law by direct experiment or calculation. Rather, he made some ingenious analogies based on careful observation of the motion of mechanical models.12 And unlike Newton, he didn’t generalise planetary motion to mathematical laws for all types of motion—and consequently he didn’t deduce, as Newton did, the conservation of what we now call kinetic and potential energy. (Leibniz, too, pioneered the notion of conservation of energy.) Nor did Hooke use his idea of gravity to estimate the oblate shape of the Earth or to explain the tides, the precession of the equinoxes, or any of the other mysteries that Newton explained. So, when Hooke accused him of plagiarism, Newton was furious: while Hooke’s mathematical ability was greater than has long been assumed, it was simply no match for Newton’s. No wonder Newton fumed to Edmond Halley that if Hooke could claim to be the inventor of the theory of gravity, then the mathematicians that “find out, settle and do all the business must content themselves with being nothing but dry calculators and drudges.” He had a point. It is interesting that in copyright law today, it is forms of expression rather than ideas that are protected. And although recently there has been an important move among maths educators to emphasise the role of calculation in the history and application of mathematics, many modern mathematicians are in tune with Newton, seeing beauty and creativity in their work, not just computations and applications.13

牛顿并不反对计算。他的大部分研究都致力于计算,因为并不总是能够找到方程的精确代数解,更不用说解决更复杂的问题了。所以你必须用数字来处理它,代入数字直到得到一个足够准确的答案——这有点让人想起古老的穷举法。卡尔达诺就是用这种方法得到x = 4 的解,但牛顿开发了不止一种而是几种系统的数值方法。

Not that Newton was averse to computation. Much of his research was devoted to it, for it is not always possible to find exact algebraic solutions of equations, let alone to solve more complex problems. So you have to come at it numerically, plugging in numbers until you get a suitably accurate answer—somewhat reminiscent of the ancient method of exhaustion. Cardano did this to get to his solution x = 4, but Newton developed not one but several systematic numerical approaches.

至于《自然哲学的数学原理》,在它首次出版五十年后,仍有一些人认为牛顿的方法过于数学化——方程式无法像机械模型那样提供因果解释。在法国,牛顿的早期支持者包括埃米莉·杜·夏特莱和她的搭档、煽动性剧作家伏尔泰。18 世纪 30 年代,他们合作出版了一本畅销书,旨在将牛顿的引力和光理论带给更广泛的读者。 (除了相信光是物质粒子之外,牛顿还做了很多细致的实验,包括证明普通阳光是由彩虹中的所有颜色组成的实验。杜·沙特莱进一步阐述了这一观点,她认为不同的颜色与不同的热量有关——这在当时是一个前沿的想法,她通过将床单染成彩虹色并计算每种颜色干燥所需的时间来进行测试:紫色最先干燥,然后是其他颜色,红色干燥时间最长。这是一个漂亮的 DIY 实验,实验表明紫色是最冷的颜色,红色是最暖的颜色,尽管这需要后来的科学家来证实和解释。14 杜·沙特莱还写了一本书,比较了莱布尼茨和牛顿的思想。但她留下的最持久的遗产是她将《自然哲学的数学原理》从拉丁文翻译成法文——这是第一部将《自然哲学的数学原理》翻译成英语以外的日常语言的译本。该译本非常出色,至今仍是权威的法文版。相比之下,最初的英文译本早已被超越。(现代版由 I. Bernard Cohen 和 Ann Whitman 于 1999 年出版。他们发现 du Châtelet 的译本特别有助于澄清一些使用 17 世纪术语的晦涩段落,这些术语后来已被现代化。)

As for Principia, fifty years after its first publication there were still some who thought Newton’s approach was too mathematical—that an equation couldn’t provide a causal explanation the way a mechanical model could. Among Newton’s early champions in France were Émilie du Châtelet and her partner, the provocative playwright Voltaire. In the 1730s, they collaborated on a popular book aimed at bringing Newton’s theories of gravity and light to a wider audience. (Aside from believing that light was a material particle, Newton did many meticulous experiments, including those showing that ordinary sunlight is made up of all the colours of the rainbow. Du Châtelet expanded on this by suggesting that the different colours are associated with different amounts of heat—a cutting-edge idea at the time, which she tested by dyeing a bedsheet in rainbow colours and timing how long it took each colour to dry: violet was first, and then the other colours in order, with red taking the longest. It was a beautiful DIY experiment, suggesting that violet is the coolest colour and red the warmest, although it would take later scientists to confirm and explain it.14) Du Châtelet also wrote a book comparing the ideas of Leibniz and Newton. But her most enduring legacy is her translation of Principia from Latin to French—the first such translation into an everyday language other than English. It was so good that it is still the definitive French version. By contrast, the original English translation has long since been surpassed. (The modern version is by I. Bernard Cohen and Ann Whitman, published in 1999. They found du Châtelet’s translation particularly helpful in clarifying some obscure passages using seventeenth-century technical terms that have since been modernised.)

杜·夏特莱直到三十岁左右才开始接触数学——作为一个女孩,她没有接受正规教育(就像一个世纪后的索菲·杰曼和玛丽·萨默维尔一样),并在十八岁时嫁给了一位合适的贵族——杜·夏特莱侯爵。她与知识发现和伏尔泰交织在一起的爱情故事非常浪漫,但她作为一名数学家的成长经历也真正鼓舞人心。她在香槟的城堡里建起了房子——以及一个最先进的图书馆和科学实验室——她和伏尔泰吸引了来自四面八方的作家和学者。 (实际上,这是她丈夫的城堡,但他是个正派人,接受了伏尔泰在妻子心中的地位。毕竟,他的婚姻是包办婚姻,不是爱情婚姻,而且他经常外出打仗,就像贵族应该做的那样,还经常和自己的情人调情。此外,伏尔泰还支付了这座古城堡大部分必要的翻修费用。)游客们陶醉于热闹的谈话和伏尔泰专门为他们写的小剧本,以及安静的研究空间。正是这样的一位游客启发了夏特莱翻译《自然哲学的数学原理》,因为她已经成为最新数学物理学的公认权威。

Du Châtelet hadn’t come to mathematics until she was about thirty—as a girl, she’d been denied formal education (just like Sophie Germain and Mary Somerville a century later) and had been married off at eighteen to a suitable aristocrat, the Marquis du Châtelet. The story of her intertwined love affairs with intellectual discovery and with Voltaire is wonderfully romantic, but her development as a mathematician in her own right is truly inspiring. Setting up house—and a state-of-the-art library and scientific laboratory—in her château in Champagne, she and Voltaire attracted writers and scholars from far and wide. (Actually, it was her husband’s château, but he was a decent man who had come to accept Voltaire’s place in his wife’s heart. After all, his had been an arranged marriage, not a love match, and he was often away fighting wars, as aristocrats were expected to do, and dallying with his own lovers. Besides, Voltaire paid for much of the necessary renovation of the old castle.) Visitors reveled in the lively conversation and the little plays written especially for them by Voltaire, as well as the peaceful space for research. It was one such visitor who inspired du Châtelet to translate Principia, for she had become a recognised authority on the latest mathematical physics.

尽管如此,在女性没有知识分子地位的时代,对于一个主要靠自学的女性来说,这仍然是一项令人难以置信的事业。起初,她不得不自学数学——尽管后来她聘请了两位法国最优秀的数学家皮埃尔-路易·莫罗·莫佩尔蒂和亚历克西斯-克劳德·克莱罗作为导师。伏尔泰也参加了一些课程,但他很快就屈服于她在数学上的优越性,开玩笑地称她为“夏特莱的牛顿夫人”。

Still, it was an incredible undertaking for a mostly self-taught woman at a time when women had no intellectual standing. Initially she had had to teach herself maths—although later she employed as tutors two of France’s best mathematicians, Pierre-Louis Moreau Maupertuis and Alexis-Claude Clairaut. Voltaire had joined some of the lessons, but he soon deferred to her mathematical superiority, playfully dubbing her “Madame Newton du Châtelet.”

除了翻译牛顿的 500 多页原文外,夏特莱夫人还提供了一份 180 页的附录,其中包括 110 页的牛顿以来引力理论发展的概述(包括她的朋友莫佩尔蒂和克莱罗的研究),以及 70 页的附加数学知识,旨在阐明牛顿的工作原理。众所周知,牛顿选择使用创新的几何构造而不是符号微积分计算来证明他的大部分定理(即使在使用微积分时也是如此)。15也许霍布斯的“符号疤痕”仍然引起共鸣。问题是,牛顿的证明如此巧妙和独特,以至于它们不能轻易地被推广——这意味着他的运动思想不能轻易地扩展到新的应用和见解。一代又一代的学者都对他感到失望:他发明了微积分,那么为什么不更透明地使用它呢?显然,他认为几何是当时最严格、最直观的数学形式。

In addition to translating Newton’s 500-odd pages, du Châtelet provided a 180-page appendix. This included a 110-page outline of developments in gravitational theory since Newton—including those by her friends Maupertuis and Clairaut—and 70 pages of additional mathematics, designed to clarify Newton’s working. He had famously chosen to present the proofs of most of his theorems using innovative geometrical constructions rather than symbolic calculus calculations—even when he was using calculus.15 Perhaps Hobbes’s “scab of symbols” still resonated. Trouble was, Newton’s proofs were so clever and idiosyncratic that they couldn’t readily be generalised—which meant that his ideas of motion couldn’t readily be extended to new applications and insights. Generations of scholars have been frustrated with him: He invented calculus, so why didn’t he use it more transparently? Evidently, he thought geometry was the most rigorous and intuitive form of mathematics at that time.

例如,将速度等导数以几何学的形式想象成三角形斜边的斜率,其边代表距离和时间的无穷小变化,这当然非常有帮助。图 2.4显示了这个想法,以及莱布尼茨的代数符号——我们熟悉的dd—反映了这种几何的、直观的解释。

For example, it is certainly very helpful to imagine a derivative, such as speed, geometrically—as the gradient of the hypotenuse of a triangle whose sides represent the infinitesimal changes in distance and time. Figure 2.4 shows the idea, and Leibnizian algebraic symbolism—the familiar dydx—reflects this geometrical, intuitive interpretation.

然而,在现代导数概念中,莱布尼茨符号有点误导,因为dd根本不是比率或分数,不是dy除以dx的数。相反,dd是作用于函数 y ( x ) 的算符——用现代语言来说;它只是意味着符号dd指导你通过求导数来“操作” y ( x )。然而,莱布尼茨的导数符号可以像处理普通分数一样进行操作。一个经典的例子是“链式法则”,你可以写出如下方程dddd=dd,它确实看起来就像您只是取消了dx一样。

In the modern conception of a derivative, however, Leibnizian notation is a little misleading, because dydx is not a ratio or fraction at all, in the sense of a number dy divided by a number dx. Rather, ddx is an operator acting on a function y(x)—to use modern language; it just means that the symbol ddx directs you to “operate” on y(x) by taking its derivative. Nevertheless, the Leibnizian symbols for derivatives can be manipulated as if they really are ordinary fractions. A classic case is the “chain rule,” where you can write equations such as dydxdxdt=dydt, which does look for all the world as if you simply canceled the dx’s.

牛顿对严谨性更加谨慎,他没有这样的透明的象征主义。例如,用莱布尼茨符号,你可以写d=ddd=d,这很容易表明,以速度v运动的物体在dt瞬间行进了微小的距离dx。相比之下,在牛顿的符号中,这将是vo = ẋo,因此从符号上看,物理解释并不那么明显。无论如何,他似乎认为,如果他坚持字面上的几何构造,《数学原理》会更容易理解:了解引力理论会引起争议,他不想在微分消失和芝诺悖论仍未解决的情况下过多地研究新微积分。当他需要微积分时,他倾向于将其写成图下的面积和几何比率,就像这样图 2.4中,尽管他有时确实使用代数来解释他的算法和图表(正如我在前面的尾注中所示,即第 15 号)。

Newton was more cautious about rigour, and he didn’t have such a transparent symbolism. For instance, with Leibnizian symbols you can write vdt=dxdtdt=dx, which readily shows that an object moving at speed v travels a tiny distance dx in the instant dt. By contrast, in Newton’s notation this would be vo = ẋo, so a physical interpretation is not so obvious from the symbolism. At any rate, he seemed to think that Principia would be more accessible if he stuck to literal geometric constructions: knowing the theory of gravity would be controversial, he didn’t want to make too much of the new calculus with its disappearing differentials and with Zeno’s paradoxes still unresolved. When he needed calculus, he tended to write it in terms of areas under graphs and geometric ratios like that in figure 2.4, although he did sometimes use algebra to explain his algorithms and diagrams (as I illustrated in the previous endnote, i.e. no. 15).

图像

图 2.4 . PQ 线的斜率为ΔΔ;它近似于 P 和 Q 之间图的梯度。更技术化、更现代地讲,我们需要极限:为了得到 P 处的精确梯度——从而找到 P 处的导数——你需要将点 Q 越来越靠近 P,直到 Δ x和 Δ y无穷小。然后,当 Δ x → 0 时,图在 P 处的导数就是ΔΔ取极限时,莱布尼茨符号dd取代ΔΔ。如果你有距离x与时间t的图表(而不是像这里显示的yx 的图表),那么梯度现在将是dd,代表速度v——单位时间的距离。

FIGURE 2.4. The slope of the line PQ is ΔyΔx; it approximates the gradient of the graph between P and Q. To be more technical—and modern—about it, we need limits: To get the exact gradient at P—and thereby find the derivative at P—you move the point Q closer and closer to P until Δx and Δy are infinitesimally small. Then the derivative of the graph at P is the limit, as Δx → 0, of ΔyΔx. When the limit is taken, Leibnizian notation dydx replaces ΔyΔx. If you had a graph of distance x versus time t (instead of y versus x as shown here), the gradient, which would now be dxdt, would represent the speed v—the distance per time.

半个世纪后,最初得益于瑞士数学家约翰·伯努利(或让·伯努利),微积分符号系统至少在欧洲大陆变得更加被接受。在莱布尼茨与牛顿之间臭名昭著的优先权之争中,伯努利是莱布尼茨最热心的辩护人,后来他与杜·夏特莱的圈子保持着联系。在《数学原理》(法语版《数学原理》)的附录中,杜·夏特莱用代数微积分重新阐述了牛顿的几十个证明。而且,英国人无疑认为这是一种不受欢迎的讽刺,她用的是莱布尼茨的符号系统,而不是牛顿的。16

Half a century later, thanks initially to the formidable Swiss mathematician Johann (or Jean) Bernoulli, calculus symbolism had become more acceptable—on the continent, at least. Bernoulli had been Leibniz’s most avid defender during the infamous priority dispute with Newton, and later he’d corresponded with du Châtelet’s circle. In the appendix to Principes Mathématiques (her French Principia), du Châtelet reworked dozens of Newton’s proofs in terms of algebraic calculus. And, in what the British no doubt saw as an unwelcome irony, she did it using Leibniz’s symbolism, not Newton’s.16

然而,通过将牛顿的几何微积分转化为莱布尼茨符号,杜·夏特莱和伯努利及其学生帮助将牛顿的愿景完全展现出来。正是由于他们以及牛顿,微积分才在物理学和技术领域中发挥了如此重要的作用。

Yet, by translating Newton’s geometrical calculus into Leibnizian symbols, du Châtelet and Bernoulli and his students helped bring Newton’s vision into full view. It is thanks to them, as well as to Newton, that calculus has proved so important in physics and technology.

• • •

• • •

然而,在本书的其余部分,我们将探讨矢量和张量微积分,以及矢量和张量本身的思想和应用。我刚才谈到了速度作为导数,但正如我在序言中提到的,“速度”是一个更精确的量:它给出了运动的方向和速度,因此它包含有关两个不同事物的信息。这就是汉密尔顿所说的“矢量”。速度本身,就像任何“标量”或数字一样,只代表一个属性。

It is vector and tensor calculus, however, that we’ll explore in the rest of this book, along with the ideas and applications of vectors and tensors themselves. I spoke just now about speed as a derivative, but as I mentioned in the prologue, “velocity” is a more precise quantity: it gives the direction of the motion as well as the speed, so it contains information about two different things. It is what Hamilton called a “vector.” Speed alone, like any “scalar” or number, represents just one attribute.

这样说来,向量的概念似乎非常简单。但正如前面几节所暗示的,它对数学家、物理学家和工程师的用处也取决于它的符号表示。那么,如何表示和计算向量呢?这是我们现在要开始探讨的一个主题,以及各种有趣问题的答案。例如,如果向量真的如此简单,那么汉密尔顿在布鲁姆桥的顿悟有什么重要意义呢?

When you put it like this, the concept of a vector seems very simple. As the preceding sections have suggested, though, its usefulness to mathematicians, physicists, and engineers will also hinge on its symbolic representation. So, how to represent, and calculate with, a vector? That’s a topic we’ll now begin to explore—along with answers to various intriguing questions. For instance, if vectors really are so simple, what was so important about Hamilton’s epiphany at Broome Bridge?

现在是时候更深入地探究这个故事了。

It is time to dive more deeply into the story.

(3)矢量图

(3) IDEAS FOR VECTORS

由于数学家并不以涂鸦闻名,威廉·罗文·汉密尔顿在布鲁姆桥上的涂鸦引起了很多关注。但是,这种导致这种狂热的轻微破坏行为的灵光一现的灵感,很少没有经过长期深刻的创造性思考。就向量而言,汉密尔顿花了数百年的时间,经过许多数学思想家的努力,才将所有的想法都整合在一起,最终得出了他自己深刻的结论。所以,让我从重温令人敬畏的牛顿开始这个故事的这一章。

Since mathematicians are not known for their graffiti, William Rowan Hamilton’s scratchings on Broome Bridge have attracted a lot of attention. But the kind of lightning-bolt inspiration that led to this ecstatic act of minor vandalism rarely comes without a long history of deep, creative thinking. In the case of vectors, it took hundreds of years, and many mathematical thinkers, for all the ideas to fall into place so that Hamilton could score his own deep conclusions in stone. So let me begin this chapter of the story by revisiting the redoubtable Newton.

这些天他并不总是表现得很好,但最近有人问我是否想请他共进晚餐。我已经提到过我钦佩他的天才,但我也喜欢他的人性化小细节——例如,他会把书折角,并且全神贯注于工作而忘了吃饭。还有,那棵传奇的苹果树仍然生长在他家的花园里,并在世界各地都有后代——包括我自己的大学。但更重要的是,我把这次与牛顿坐在一起的邀请当作一个重新研究《自然哲学的数学原理》的机会,我第一次不仅被它的范围和它对微积分的开创性应用所震撼,而且被它对“矢量”数量的基本性质的清晰阐述所震撼。

He doesn’t always come off well these days, but recently I was asked if I’d like to invite him to dinner. I’ve mentioned already my admiration of his genius, but I also like the little human touches—dog-earing his books, for example, and becoming so engrossed in his work that he forgot to eat. Then there’s the fact that the legendary apple tree is still growing in the garden of his family home and has descendants around the world— including at my own university. More seriously, though, I took this invitation to sit with Newton as a chance to dip back into Principia, and for the first time I was struck not just by its scope and its pioneering applications of calculus, but by its clear exposition of the fundamental nature of “vectorial” quantities.

我以前从未关注过这个问题——我只是想当然地认为向量,所以我没有意识到牛顿的处理有多么重要。我没有意识到,在为基于证据、可测试的自然物理理论制定蓝图时,他看到了定义基本物理量的必要性,例如力、速度和动量,以便它们既有方向又有大小。正如我们所见,今天的高中数学学生学会用箭头来表达这种双重性质。但矢量本身的出现却花了很长时间,甚至连牛顿都没有完全理解这个概念。

I’d never paid this any attention before—I’d simply taken vectors for granted, so I hadn’t realised how significant Newton’s treatment was. I hadn’t realised how telling it is that, in creating the blueprint for evidencebased, testable physical theories of nature, he saw the need to define fundamental physical quantities, such as force, velocity, and momentum, so that they have both direction and magnitude. Today’s high school maths students learn to express this dual nature as an arrow, as we’ve seen. But the vector itself was a surprisingly long time in coming, and not even Newton got the full concept.

他在这方面的关键见解是从力对物体的影响的角度对力进行定义。在他著名的第二运动定律中,他说,有效力与由此产生的“运动改变”成正比,而这种“改变”与力的方向相同。使用文字形式的方程,牛顿已经对“运动量”(他在第二定律中称之为“运动”)给出了仔细的定义,即“物质的量”(质量)乘以速度——因此在符号中运动量为mv,我们称之为“动量”。因此,用现代语言来说,牛顿的“运动改变”是物体动量的变化率,用矢量微积分来说,第二定律为:

His key insight in this context was his definition of force in terms of its effect on material bodies. The effective force, he said in his famous second law of motion, is proportional to the resulting “alteration of motion,” and this “alteration” is in the same direction as the force. Using word-form equations, Newton had already given a careful definition of “quantity of motion” (which he just called “motion” in the second law) as the “quantity of matter” (mass) times the velocity—so in symbols the quantity of motion is mv, which we call “momentum.” In modern language, then, Newton’s “alteration of motion” is the rate of change of the body’s momentum, and in terms of vector calculus, the second law is:

F=dd

F=dmvdt.

如果物体的质量在运动过程中没有发生可测量的变化(除非相对于测量者来说物体移动得非常快),那么这就相当于F=dd,或者说质量乘以加速度,F = m a。我用粗体字写了F、av,以表明它们是矢量,但这是事后才知道的。虽然很明显,牛顿的力和动量既有大小又有方向,但它们还不是完整的数学矢量,因为没有如何将它们相加或相乘的一般概念。你可能会认为,担心相加和相乘是一种神秘的区别,因为矢量最明显的特征是它能够同时表示大小和方向。然而,正如我们将看到的,矢量相乘开辟了一系列全新的物理应用。正是矢量的代数,被视为数学量,而不是简单的物理类似物,才使得矢量不仅适用于理论物理学,还适用于数值建模、数据分析、工程、人工智能和机器人等领域。

If the body’s mass doesn’t change measurably during the motion— which it doesn’t unless it is moving extremely fast relative to the measurer—then this is the same as F=mdvdt, or mass times acceleration, F = ma. I’ve written the F, a, and v in bold type to indicate that they are vectors, but this is hindsight. Although it was clear that Newton’s force and momentum had both magnitude and direction, they were not yet full mathematical vectors, for there was no general concept of how to add or multiply them. You might think that worrying over adding and multiplying is an arcane distinction, given that the most obvious characteristic of a vector is its ability to represent both magnitude and direction at the same time. As we’ll see, though, multiplying vectors opens up a whole new range of physical applications. And it is the algebra of vectors, treated as mathematical quantities rather than simply as physical analogues, that makes vectors applicable not only in theoretical physics, but also in fields such as numerical modeling, data analysis, engineering, AI, and robotics.

图像

图 3.1两个矢量AB相加。牛顿和其他使用平行四边形的先驱者在这里可能并没有想到要添加矢量或“线”,但我们今天的想法是:AB是结果矢量A + B的分量。平行四边形规则的一个关键特征是,此类矢量和中的两个分量作用独立,每个分量的行为都好像另一个分量不存在一样。换句话说,当你使用平行四边形规则添加两个分量时,不会改变它们的原始大小或方向,而只是平移它们以形成平行四边形的相对边。

FIGURE 3.1. Adding two vectors A and B. Newton and the other early pioneers who used the parallelogram probably did not think in terms of adding vectors or “lines” here, but this is how we think of it today: A and B are components of the resulting vector A + B. A key feature of the parallelogram rule is that the two components in a vector sum such as this act independently, in the sense that each behaves as if the other weren’t there. In other words, when you add the two components using the parallelogram rule, you don’t change their original sizes or directions, but simply translate them to form the opposite edges of the parallelogram.

与此同时,在牛顿时代,有简单加法这一实用概念,即两个矢量通过图 3.1所示的“平行四边形规则”以几何方式相加。实际上,这里的“加法”一词在矢量正式发展之前可能已经过时了——“合成”才是更准确的说法。在牛顿运动定律的前两个推论中,牛顿对这一规则做出了清晰的解释,并明确展示了它如何使对角线方向的力由另外两个“分”力合成(反之,也可以“分解”为其分力)。当然,平行四边形规则也适用于速度,牛顿在其行星和日常运动理论中对其进行了精彩的运用。

Meantime, in Newton’s day there was a practical notion of simple addition, where two vectorial quantities were added geometrically via the “parallelogram rule” shown in figure 3.1. Actually, the term “addition” here is probably anachronistic before the formal development of vectors— “composition” is a more accurate term. In the first two corollaries to his laws of motion, Newton gave a clear explanation of this rule, and showed explicitly how it enables a force in the diagonal direction to be composed from two other “component” forces (and conversely, it can be “decomposed” into its components). The parallelogram rule applies to velocities, too, of course, and Newton went on to make spectacular use of it in his theories of planetary and everyday motion.

“运动”这个概念看似简单,但正如芝诺所言,想出一个有用的定义并不容易,尤其是一个能让你提前预测物体在各种力的作用下如何运动的定义。牛顿的对手莱布尼茨在 1695 年的《动力学论文》中花费了大量篇幅试图弄清楚这一点,但他没有提到《数学原理》 。牛顿后来以牙还牙,从《数学原理》第三版中删除了对莱布尼茨微积分的认可——不幸的微积分优先权之争如火如荼地展开。莱布尼茨在《动力学论文》中对牛顿的忽视不仅仅是但这未免有些粗鲁:它表明了理解新思想是多么困难。莱布尼茨也许是他那一代最伟大的哲学家,也是一位创新的科学思想家,但就连他似乎也没有意识到,当牛顿用由此产生的动量变化来定义力时,他在运动分析方面取得了巨大的进步。(当然,后来的物理学家确实承认了这一成就,并最终将力的 SI 单位1命名为“牛顿”。)

It seems such a simple concept, “motion,” but, as Zeno showed, it’s not so easy to come up with a useful definition—especially one that allows you to predict in advance how a body will move under various forces. Newton’s rival Leibniz spent many wordy pages trying to figure it out, in his 1695 Essay on Dynamics—but he didn’t mention Principia. Newton later returned the favour by removing his acknowledgment of Leibniz’s calculus from Principia’s third edition—the unfortunate calculus priority dispute was in full swing. Leibniz’s neglect of Newton in his Essay was more than churlishness, though: it shows how difficult it can be to understand new ideas. Leibniz was perhaps the greatest philosopher of his generation, and an innovative scientific thinker, yet even he didn’t seem to realise that Newton had made a great step forward in the analysis of motion when he defined force in terms of the resulting change in momentum. (Later physicists did recognise this achievement, of course, and ultimately named the SI unit1 of force the “newton.”)

然而,有趣的是,即使是牛顿也没有提出向量代数的概念,更不用说向量微积分了。这凸显了提出新的数学概念有多么困难——也表明向量看似简单,其实是骗人的。

It is intriguing to realise, though, that not even Newton developed the idea of vector algebra, let alone vector calculus. It highlights just how hard it is to come up with new mathematical concepts—and it suggests that the seeming simplicity of vectors is deceptive.

• • •

• • •

事实上,当牛顿写作《自然哲学的数学原理》时,他已经经过了漫长的道路,集结了世界上最优秀的思想家,才得出平行四边形定律。像牛顿和莱布尼茨一样,这些思想家也一直在努力理解事物如何运动以及力如何作用。想象一下如何从头开始定义“力”,你就能体会到他们所面临的困难。牛津英语词典给出的日常定义是“力量;施加的力量或动力;巨大的努力”。这给了你大概的概念,但并不能帮助你发射通讯卫星或设计风力涡轮机。对于这类应用,你需要一个定量的矢量微积分定义。

In fact, when Newton was writing Principia, it had already been a long road, peopled with some of the best thinkers in the world, just to arrive at the parallelogram rule. Like Newton and Leibniz, these thinkers had been trying to work out how to understand how things move and how forces act. You can get a sense of the difficulty they faced by imagining how you would go about defining “force” from scratch. The everyday definition given in the Oxford English Dictionary is “power; exerted strength or impetus; intense effort.” That gives you the general idea, but it’s not going to help you launch a communications satellite or design a wind turbine. You need a quantitative, vector calculus definition for that kind of application.

当然,很明显,如果你用两倍的力推一个物体,它就会移动两倍远——这是古代哲学家很久以前就采取的量化步骤。不过,在这种情况下,两次推力的方向与产生的运动方向相同,因此方向和强度不应该成为力本身定义的一部分,这一点并不明显。但古人确实瞥见了从不同方向的两个不同“分量”组成轨迹的想法——例如在托勒密的行星运动机械模型中,行星旋转就像它们由齿轮驱动一样,齿轮也沿着轨道缓慢滚动。

Of course, it’s obvious that if you push twice as hard on an object it will move twice as far—a quantifying step that ancient philosophers had taken long ago. In this case, though, both pushes are in the same direction as the resulting motion, so it isn’t apparent that direction as well as strength should be part of the definition of force itself. But the ancients did glimpse the idea of composing a trajectory from two different “components” in different directions—such as in Ptolemy’s mechanical model of planetary motion, where the planets whirled about as if they were powered by gears that also rolled slowly along their orbits.

通过平行四边形规则将不同的运动结合起来的第一个暗示似乎是在《机械问题》中,它显然是由一位他是公元前四世纪末亚里士多德学派的成员,比托勒密早了近五百年。这是一份失传已久的手稿,直到文艺复兴时期,在中世纪伟大的阿拉伯翻译运动之后,才在欧洲重新发现——其中对力学的处理启发了塔尔塔利亚和伽利略等早期现代物理学家。他们试图找出如何从分运动中组合出移动物体的路径,例如当网球抛向空中时,斜向上运动和重力向下作用的组合。2

The first hint of combining different motions via the parallelogram rule seems to be in Questiones Mechanicae, which was apparently written by a member of Aristotle’s school at the end of the fourth century BCE, nearly five hundred years before Ptolemy. It was one of those long-lost manuscripts that were only rediscovered in Europe during the Renaissance, in the wake of the great medieval Arabic translation movement—and its treatment of mechanics proved inspirational to early modern physicists such as Tartaglia and Galileo. They were trying to work out how to compose the paths of moving objects from component motions, such as the combination of an oblique upward motion and the downward effect of gravity when a tennis ball is lobbed into the air.2

实际上,塔尔塔利亚和伽利略对网球的兴趣远不及箭和炮弹;当时的宗教和帝国战争似乎永无休止。枪支和大炮变得越来越复杂——例如,新的铸铁技术意味着炮弹现在更轻但威力更大——而且越来越明显的是,提前知道导弹的轨迹将使这些新武器能够更有效地部署。炮手手册也出版了,记录了观察到的各种射击的射程和角度,数学家开始使用这些数据试图用数学方法描述射弹的路径。

Actually, Tartaglia and Galileo were interested not so much in tennis balls as in arrows and cannonballs; such were the times, with their seemingly endless religious and imperial wars. Guns and cannon were becoming more sophisticated—for example, new cast-iron techniques meant cannonballs were now lighter but more powerful—and it was increasingly clear that knowing a missile’s trajectory in advance would enable these new weapons to be deployed more efficiently. Gunners’ manuals were being published, too, setting down records of the observed ranges and angles of various shots, and mathematicians began to use these data to try to describe the path of a projectile mathematically.

首先,在 1530 年代早期——在与卡尔达诺就三次方程展开激烈争论的几年前——塔塔利亚设法算出,如果想让炮弹飞得尽可能远,大炮应与地面呈 45° 角。如果你只是将炮弹直直地指向远处的目标,重力可能会让你无法击中目标。如果你将炮弹指向高处并希望获得最佳效果——这早已是惯常做法——那么你通常会错过目标并摧毁其他东西。但塔塔利亚遭受了良心危机,因为他知道他在研究战争武器时是在拿人们的生命开玩笑。因此,他烧毁了自己的论文,担心上帝会因他从事这种“恶毒而残酷”的工作而发怒。3

As a first step, in the early 1530s—a few years before his acrimonious quarrel with Cardano over cubics—Tartaglia managed to work out that if you want a cannonball to go as far as possible, the cannon should be pointed at an angle of 45° to the ground. If you just pointed it straight ahead at a distant target, gravity would likely catch you short. And if you pointed it high and hoped for the best—which had long been the usual practice— then you’d often miss your target and destroy something else instead. But Tartaglia suffered a crisis of conscience, for he knew he was dallying with people’s lives while he worked on weapons of war. So he burned his papers, fearing God’s wrath for engaging in such “vituperative and cruel” work.3

然而,那是一个可怕的时代。塔尔塔利亚在意大利北部长大,小时候被一名法国士兵的军刀刺伤,伤势严重。当时他十二三岁,和母亲、妹妹躲在一座教堂里,当时法国人正在攻打他们的城镇,但那把锋利的剑尼科洛的嘴和下巴被一枪打中。这枪的伤痕留在了他的名字塔尔塔利亚身上——他的真实姓氏似乎是丰塔纳,但在上颚受伤后,他得到了“塔尔塔利亚”的绰号,即口吃者。尽管如此,尽管他缺乏教育,他还是成为了十六世纪初最优秀的数学家之一。至于他在弹道学方面的工作,他很快决定,作为一个虔诚的基督徒,他最好和他的赞助人谈论它,因为伊斯兰奥斯曼帝国正在进一步向基督教欧洲扩张。它已经扩展到希腊和土耳其的大部分地区,最北延伸到贝尔格莱德和索非亚,但到了 1530 年代,苏莱曼皇帝的军队入侵了匈牙利,并与奥地利发生了冲突。奥地利哈布斯堡王朝也在扩大自己的影响力,法国和奥斯曼帝国结成联盟对抗他们——就这样,战争似乎无处不在。

These were fearful times, however. As a child growing up in northern Italy, Tartaglia had suffered a terrible wound from a French soldier’s saber. He was twelve or thirteen and had been hiding with his mother and sister in a church as the French stormed their town, but the slashing sword caught young Niccolò across the mouth and jaw. The wound’s legacy lives on in his name, Tartaglia—his actual surname seems to have been Fontana, but after the injury to his palate he acquired the nickname “Tartaglia,” the Stammerer. Despite all this, and despite his lack of education, he became one of the best mathematicians of the early sixteenth century. As for his work on ballistics, he soon decided that as a good Christian he’d better talk about it to his patron, for the Islamic Ottoman Empire was expanding ever further into Christian Europe. It already extended through much of Greece and Turkey, and as far north as Belgrade and Sofia, but by the 1530s Emperor Süleyman’s troops had invaded Hungary and were skirmishing with Austria. The Austrian Habsburgs were extending their own influence, too, and the French and the Ottomans formed an alliance against them— and so it went: war seemed to be everywhere.

于是塔尔塔利亚继续研究这个问题。他按照建议将大炮倾斜 45° 度,设计了一种象限来测量这种角度。他还试图构建一个理论弹道。直观上看,如果忽略空气阻力(对于炮弹来说这是一个足够现实的假设,但对于羽毛来说则不然),这样的路径仅由两个不同的分量构成:一个在发射方向上,一个在重力方向上。但塔尔塔利亚的尝试没有成功。首先,他的分量不是明确定义的力或速度,而只是直观的“运动”——毕竟,他写作的时间比牛顿严谨、开创性的运动定律早 150 年。更重要的是,塔尔塔利亚从未以矢量方式添加(或“构建”)这些“运动”。在对重新发现的《机械问题》的评论中,他只是忽略了平行四边形规则——像他的大多数同时代人一样,他不理解它的重要性。4

So Tartaglia kept working on the subject. He followed up his advice about angling cannon at 45° by designing a kind of quadrant for measuring such angles. He also tried to compose a theoretical ballistic trajectory. It was intuitively evident that if you ignored air resistance—a realistic enough assumption for a cannonball although not for a feather—such a path was built from just two different components: one in the direction the shot was fired, and one in the direction of gravity. But Tartaglia’s attempt wasn’t successful. First, his components were not clearly defined forces or velocities but just intuitive “motions”—after all, he was writing 150 years before Newton’s careful, groundbreaking laws of motion. What’s more, Tartaglia never added (or “composed”) these “motions” vectorially. In his commentary on the rediscovered Questiones Mechanicae, he’d simply ignored the parallelogram rule—like most his contemporaries, he didn’t understand its significance.4

其次,他没有独立分量的概念,即每个分量都像其他分量不存在一样起作用。这是平行四边形规则的关键,如图 3.1所示。举个例子,如果你在这个分析中去掉重力,只让其中一个分量起作用——大炮发射方向的分量——你应该推断出炮弹会沿着初始方向以相同的速度继续移动。这是牛顿第一运动定律,尽管其他人,尤其是伽利略和哈里奥,在他之前就已经直觉地发现了它。

Second, he didn’t have the idea of independent components, each one acting as if the other weren’t there. This is key to the parallelogram rule, as you can see in figure 3.1. An example of what this means is that if you took away gravity in this analysis so that only one of the components acted—the one in the direction the cannon was fired—you should deduce that the ball would keep moving at the same speed along that initial direction. This is Newton’s first law of motion, although others, notably Galileo and Harriot, had intuited it before him.

图像

图 3.2 . 塔尔塔利亚等早期理论家对抛射物路径概念化的草图。

FIGURE 3.2. Sketch of how early theoreticians such as Tartaglia conceptualised the path of a projectile.

然而,就像亚里士多德之后的几乎所有人一样,塔尔塔利亚认为物体不可能有两个独立的运动分量同时作用。所以他没有想到通过平行四边形规则将这两个分量结合起来,从而得到最终的观测运动。如图 3.2所示,他的轨迹由一个初始的倾斜分量一个向下的分量组成。他并不认为两者从一开始就是一起作用的,所以炮弹的路径实际上是弯曲的,而不是直线的,因为重力已经将它拉下了。相反,在轨迹的中间,他有一个弯曲的部分,代表当重力“突然”克服了最初的向外力时炮弹的路线改变,然后炮弹垂直下落。

Tartaglia, however—like almost everyone since Aristotle—believed it was not possible for a body to have two independent components of motion acting at the same time. So he didn’t have the idea of combining the two components, via the parallelogram rule, to give the resulting observed motion. As you can see in figure 3.2, he composed his trajectory from an initial oblique component followed by a downward one. He didn’t have the idea that both were acting together from the outset, so that the ball’s path actually starts out curved, not straight, because gravity is already pulling it down. Instead, in the middle of his trajectory, he has a curved section representing the cannonball’s changing course when gravity “suddenly” overcomes the initial outward force, and then the ball drops straight down.

虽然塔塔利亚的著作是首次对弹道学进行重大研究,但他分析中的缺陷表明,没有人真正知道引力是如何运作的,更不用说理解矢量了。半个世纪后,伽利略和哈里奥特出现了。战争仍在肆虐,但现在新教徒和天主教徒之间的紧张关系带来了新的动乱。然而,人们对“战争艺术”也产生了一种智力和审美上的迷恋。战争游戏是哈里奥特的第二位赞助人诺森伯兰伯爵等人的一种娱乐消遣;他曾在军队服役一段时间,对军事战略如此着迷,以至于他花了无数的时间玩一种精心设计的棋盘游戏,其中有 140 名黄铜士兵,每个士兵都拿着一根铁丝长矛,还有 320 名铅士兵拿着小火枪。战争也是艺术的主题:欧洲大陆的文艺复兴时期艺术家,如乔治奥·瓦萨里和阿尔布雷希特·阿尔特多弗一直在描绘鸟瞰式的战争场景,而现在,在英国,威廉·莎士比亚和克里斯托弗·马洛通过强有力的语言和巧妙的舞台表演,在《亨利五世》《帖木儿》等戏剧中唤起了人们对大规模战争的回忆。《帖木儿》戏剧化地描述了十四世纪鞑靼战士帖木儿的可怕征服。它或许让人想起了苏莱曼最近的功绩,尽管马洛是在更直接的危险——西班牙无敌舰队的阴影下创作的,西班牙无敌舰队因各种长期酝酿的宗教、地缘政治和经济原因于 1588 年袭击了英国。有证据表明马洛认识哈里奥特——有人推测他们讨论过“战争艺术”,马洛是诗人和剧作家,哈里奥特是研究弹道的数学家。5

While Tartaglia’s was the first significant foray into the study of ballistics, the flaw in his analysis suggests that no one really knew how gravity operated, let alone understood vectors. Enter Galileo, and Harriot, half a century later. War was still raging, but now the tension between Protestants and Catholics was bringing new unrest. Yet there was an intellectual and aesthetic fascination with “the art of war,” too. War games were an entertaining pastime for people such as Harriot’s second patron, the Earl of Northumberland; he’d served briefly in the military and was so fascinated by military strategy that he spent countless hours playing an elaborate board game with 140 brass soldiers, each with a wire pike, and 320 lead soldiers carrying tiny muskets. War was a subject of art, too: continental Renaissance artists such as Georgio Vasari and Albrecht Altdorfer had been painting bird’s-eye battle scenes, and now, in England, William Shakespeare and Christopher Marlowe were evoking huge battles through powerful language and clever staging, in plays such as Henry V and Tamburlaine. Tamburlaine dramatises the terrifying conquests of the fourteenthcentury Tartar warrior Tamerlane. It was a reminder, perhaps, of the more recent exploits of Süleyman, although Marlowe wrote it in the shadow of a more immediate danger, the Spanish Armada, which for various longsimmering religious, geopolitical, and economic reasons attacked England in 1588. There is some evidence that Marlowe knew Harriot—and it has been conjectured that they discussed the “art of war,” Marlowe as a poet and playwright, and Harriot as a mathematician working on ballistics.5

没有证据表明伽利略和哈里奥特彼此认识,但他们对当时最前沿的科学有着共同的兴趣——不仅仅是在弹道学方面,而且在所有受重力影响的运动方面。结果,他们独立发现了自由落体的一般规律——在日常情况下,如果消除空气阻力,所有物体,无论其重量和大小,都会以相同的速度下落。但对于塔尔塔利亚来说,炮弹比网球下落得更快,所以它们的抛射轨迹形状不同。但哈里奥特和伽利略表明,在真空中,所有抛射物​​的轨迹形状相同:抛物线。如果你倾斜一根喷出水流的软管,你可以很容易地看到这个形状,因为每一滴水滴都描绘出抛射物的形状。证明它是一条抛物线,有形式为y = −ax2 + bx + c独特方程,则是另一回事。但是,如果你在高中或大学学习过数学或物理(或者如果你提前看过图 8.1),你就会知道,使用牛顿定律、向量和微积分规则,只需十几行代码就可以完成。如果没有这些技巧,伽利略和哈里奥特就花了很多页纸和许多艰苦的月份来寻找适合他们数据的正确方程式。

There’s no evidence that Galileo and Harriot knew each other, but they shared the cutting-edge scientific interests of their time—and not just in ballistics, but in all motion that is affected by gravity. As a result, they independently codiscovered the general law of free fall—that in everyday situations, if you take away air resistance, all bodies, no matter their weight and size, fall at the same rate. For Tartaglia, though, a cannonball fell faster than a tennis ball, so they’d each have differently shaped projectile trajectories. But Harriot and Galileo showed that in a vacuum, all projectiles trace the same shaped path: a parabola. You can see this shape easily if you tilt a hose with a jet of water coming out, because each droplet, one after the other, traces out the shape of a projectile. Proving it’s a parabola, with a distinctive equation of the form y = −ax2 + bx + c, is another matter. But if you’ve studied maths or physics at high school or college level (or if you peek ahead at fig. 8.1), you’ll know that it can be done in a dozen lines using Newton’s laws, vectors, and the rules of calculus. Without these techniques, it took Galileo and Harriot many pages and many arduous months as they searched for the right equation to fit their data.

相比之下,你可以在图 3.2中看到,塔尔塔利亚估计的轨迹绝对不是抛物线!伽利略和哈里奥特之所以成功,是因为他们意识到分力运动确实独立起作用——这也是他们如何直观地理解牛顿第一定律的。6伽利略对抛射体的研究影响了牛顿,牛顿在《自然哲学的数学原理》中引用了伽利略的研究。但伽利略从未真正理解完全平行四边形规则。他用它计算合力的大小,但没有计算方向——大多数其他当代先驱也是如此。勒内·笛卡尔确实用这个规则来计算大小和方向,但他认为它们是独立的量,而不是单个量(如速度或力)的独立方面。所以,你可以看到为什么矢量比我们后见之明和习惯所能意识到的要多得多!7

By contrast, you can see in figure 3.2 that Tartaglia’s estimated trajectory was definitely not a parabola! Galileo and Harriot succeeded because they realised that component motions do act independently—which is also how they came to intuit Newton’s first law.6 Galileo’s work on projectiles influenced Newton, who cited it in Principia. But Galileo never quite got to the full parallelogram rule. He used it to calculate the magnitudes of his resultant forces, but not the directions—and the same was true for most other contemporary pioneers. René Descartes did use the rule to calculate magnitudes and directions, but he thought of them as separate quantities, not as separate aspects of a single quantity, such as velocity or force. So, you can see why there is more to vectors than we, with our hindsight and habit, might realise!7

不过,哈里奥特在 1619 年对碰撞力学的分析中接近了这个想法。除了战争和战争游戏,台球游戏也风靡一时——至少对哈里奥特的赞助人诺森伯兰这样的富裕贵族来说是这样。不幸的是,伯爵很快就被关进了伦敦塔,成为詹姆斯一世国王的囚徒——他和哈里奥特在 1605 年火药阴谋被挫败时被捕,因为他们在前一天不知情的情况下与其中一名阴谋者共进晚餐。(阴谋者是心怀不满的天主教徒,对新教国王及其议会的统治感到不满,其中一名恰好是诺森伯兰的表弟。)哈里奥特在经历了几个可怕的星期后被释放,但这位无辜的伯爵就没那么幸运了:他在接下来的 16 年里都被关在监狱里。他设法通过玩战争游戏来消磨一些时间——并在塔的场地上建造了一条保龄球馆。但他也花时间思考科学,为了激发他对保龄球和台球的兴趣,他请哈里奥特解释碰撞力学。

Harriot came close to the idea, though, in his 1619 analysis of the mechanics of collisions. As well as war and war games, the game of billiards was in the air—at least for wealthy aristocrats such as Harriot’s patron Northumberland. Unfortunately, the earl soon landed in the Tower of London, a prisoner of King James I—both he and Harriot were arrested in 1605 when the Gunpowder Plot was foiled, because they had unknowingly dined with one of the Plotters just the day before. (The Plotters were disaffected Catholics chafing under the rule of the Protestant king and his parliament, and one of them happened to be Northumberland’s cousin.) Harriot was released after a few terrifying weeks, but the innocent earl was not so lucky: he spent the next sixteen years in prison. He managed to while away some of the time playing war games—and by having a bowling alley constructed in the Tower’s grounds. But he also spent his time thinking about science, and to build on his interest in bowls and billiards he asked Harriot to explain the mechanics of collisions.

哈里奥特的论文是对该主题的首次详细分析,也是牛顿之前最复杂的矢量分析之一。我在图 3.3中重现了哈里奥特的矢量图。球aA的质量由圆圈的大小表示,它们从左侧相向而行,在中间(在bB 处)相撞,然后反弹到虚线圆圈cC表示的位置。我在尾注8中解释了他的计算,但只要看一下哈里奥特的图表,你就可以看出他使用平行四边形来分析他在此处展示的两种碰撞类型。尽管他的术语很过时——他谈论运动和动力而不是速度和力——但他掌握了现代平行四边形规则的精髓:大小和方向都是指定的,他有独立的部分,每个部分都“好像”另一个部分不存在一样,它们结合起来产生合力或合力。唯一的问题是,他没有发表它。9

Harriot’s paper was the first detailed analysis of the topic—and one of the most sophisticated vectorial analyses before Newton. I’ve reproduced Harriot’s vectorial diagram in figure 3.3. The masses of the balls a and A are indicated by the size of his circles, and they move toward each other from the left, collide in the middle (at b and B), and bounce off to the positions denoted by the dotted circles c and C. I’ve explained his calculations in the endnote,8 but you can see just by looking at Harriot’s diagram that he used parallelograms to analyse the two types of collision he’s shown here. And although his terminology is old-fashioned—he speaks of motion and impetus instead of velocity and force—he has the essence of the modern parallelogram rule: both magnitudes and directions are specified, and he has independent components, each acting “as if ” the other weren’t there, which combine to give the resultant motion or force. The only trouble is, he didn’t publish it.9

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图 3.3。哈里奥特使用平行四边形规则分析碰撞力学。手稿:西萨塞克斯档案馆,PHA HMC 241 第 VIa 卷,f23r。由 Rt. Hon. Lord Egremont 提供,并感谢西萨塞克斯档案馆的县档案保管员。

FIGURE 3.3. Harriot’s use of the parallelogram rule to analyse the mechanics of collisions. The manuscript: West Sussex Record Office, PHA HMC 241 Vol VIa, f23r. By courtesy of the Rt. Hon. Lord Egremont, and with acknowledgments to the County Archivist, West Sussex Record Office.

不过,他的同胞约翰·沃利斯可能见过它——他肯定看过哈里奥特的一些手稿,以及他死后出版的代数教科书《实践》。正如我所提到的,沃利斯尤其受到哈里奥特的完全符号代数符号的启发,但他还探讨了碰撞的主题——牛顿认为他在这方面的工作是《自然哲学的数学原理》的另一个重要影响。沃利斯在 1671 年的一篇力学论文中清楚地阐述了平行四边形规则,在 1687 年《自然哲学的数学原理》出版之前的几十年里,其他几个人也这样做了——包括多才多艺的费马。(和哈里奥特的代数先驱韦达一样,费马的日常工作是法律,而不是数学。)10

His countryman John Wallis may have seen it, though—he’d certainly seen some of Harriot’s manuscripts, as well as his posthumous algebra textbook Praxis. Wallis was inspired especially by Harriot’s fully symbolic algebraic notation, as I’ve mentioned, but he also tackled the topic of collisions—Newton cited his work on this as another important influence on Principia. Wallis clearly stated the parallelogram rule in a 1671 paper on mechanics, and, in the decades leading up to 1687 when Principia was published, several others did so, too—including the remarkably versatile Fermat. (Like Harriot’s algebraic forerunner Viète, Fermat’s day job was in law, not mathematics.)10

你可能会认为,随着《数学原理》中平行四边形规则的成熟,以及牛顿对力、加速度和速度的矢量定义,正是他引发了对完整的数学矢量分析的探索——但事实并非如此,至少不是直接的。他的工作后来确实有所贡献,展示了矢量在物理学中的用处,但那是在第一个矢量代数从一个令人惊讶的、完全不同的方向出现之后。

You might think that, with the parallelogram rule coming of age in Principia, and with Newton’s vectorial definitions of force, acceleration, and velocity, it was he who sparked the search for a complete, mathematical vector analysis—but it wasn’t, at least not directly. His work did contribute later, by showing how useful vectors would be in physics, but that was after the first vector algebra had appeared from a surprising and completely different direction.

故事的惊人转折

A SURPRISING TWIST IN THE TALE

你或许可以从汉密尔顿的涂鸦中猜出这个令人惊讶的方向。它包含虚数i ,当你试图解一个方程,例如x 2 + 1 = 0时,它就会出现。更复杂的方程会给出一个更奇怪的量,即实数和虚数的混合,例如拉斐尔·邦贝利的2+111,我们在第 1 章中也曾见过。我还提到,笛卡尔在邦贝利之后大约一个世纪创造了“虚数”一词——然后,在 1777 年,也就是笛卡尔之后一个多世纪,瑞士数学家莱昂哈德·欧拉创造了现代符号i来表示1

You can perhaps guess this surprising direction from Hamilton’s graffiti. It contains the imaginary number i, which comes up when you try to solve an equation such as x2 + 1 = 0. More complicated equations give an even stranger quantity, a mixture of a real and an imaginary number, such as Rafael Bombelli’s 2+111, which we also met in chapter 1. I mentioned, too, that Descartes had coined the term “imaginary number,” about a century after Bombelli—and then, in 1777, more than a century after Descartes, the Swiss mathematician Leonhard Euler created the modern symbol i to denote 1.

欧拉还给了我们指数(或欧拉)数的符号e——我在下面对其进行了定义——他还推广了威廉·琼斯的符号π。但就像琼斯的π一样,欧拉的i直到几十年后另一位杰出数学家——德国数学家卡尔·弗里德里希·高斯——才真正流行起来。高斯还创造了“复数”一词,用于表示实数和虚数的混合,我们今天将其写为a + iba + bi —— i的书写顺序无关紧要。但“复数”一词也没有立即流行起来。所有这些都表明创新思想的演变很缓慢——如果你想获得一些迟来的荣誉,发表论文是多么重要!

Euler also gave us the symbol e for the exponential (or Euler) number—which I’ve defined below—and he popularised William Jones’s symbol π. But just as with Jones’s π, Euler’s i didn’t really take off until another standout mathematician used it several decades later—the German Carl Friedrich Gauss. Gauss also coined the term “complex number,” for the mix of real and imaginary numbers we write today as a + ib or a + bi— the order in which you write the i doesn’t matter. But the term “complex” didn’t catch on straightaway, either. All of which goes to show the slow evolution of innovative ideas—and how crucial it is to publish if you want to get some belated credit!

顺便说一句,高斯还是自学成才的女性先驱索菲·热尔曼的不知情的导师。和他那个时代的大多数男人一样,高斯认为女性无法掌握高等数学,而热尔曼曾谨慎地用勒布朗先生的笔名给他写信,谈到了证明费马大定理的想法。当她最终透露自己是女性时,他感到惊讶,也非常钦佩。

Incidentally, Gauss was also the unknowing mentor of the self-taught female trailblazer Sophie Germain. Like most men of his time, Gauss believed women were incapable of higher mathematics, and Germain had prudently written to him—about an idea for proving Fermat’s last theorem—using the pseudonym Monsieur Le Blanc. When she finally revealed that she was a woman, he was amazed, and very impressed.

但是i和向量有什么关系呢?它取决于如何表示信息的问题,以及遵循明确数学规则的哪种数值结构有资格成为数学研究的对象的相关问题。

But what has i to do with vectors? It hinges on the question of how to represent information—and the related question of what kind of numerical constructs qualify as objects of mathematical study, by obeying clear mathematical rules.

就i而言,如果数学家们要认真对待这个数字,他们该如何思考呢?他们花了很长时间才将数轴延伸到零的左边来表示负数——这一看似简单的步骤使得人们能够将负数简单地视为正数的对应物,并且同样真实。沃利斯迈出了这一重大的一步——尽管他也认为负数必须大于无穷大,这表明即使是最优秀的数学家也很难掌握负数(更不用说无穷大了)。沃利斯也是第一个尝试对复数做类似的事情的人,但他没有成功。11

In the case of i, if mathematicians were going to take this number seriously, how were they to think about it? It had taken long enough to extend the number line to the left of zero to represent negative numbers—a seemingly simple step that made it possible to think of negative numbers simply as counterparts to positive ones, and just as real. It was Wallis who took this momentous step—although he also thought that negative numbers must be larger than infinity, which shows how difficult it was for even the best mathematicians to get a handle on negative numbers (let alone infinity). It was also Wallis who first tried to do something similar for complex numbers, but he wasn’t successful.11

这个表示问题又花了一个世纪才得到解决,并无意中为发现向量开辟了道路。与此同时,数学家需要更多地了解复数的数学性质。当然,它们是作为二次和三次方程的解而出现的,但它们在数学中是否发挥了更广泛的作用——这个角色使得必须找到一种方法将它们纳入扩展的数字体系中?

It would take another century for this representation problem to be solved, and to inadvertently open the way for the discovery of vectors. Meantime, mathematicians needed to find out more about the mathematical nature of complex numbers. Sure, they came up as solutions of quadratic and cubic equations, but did they play a broader role in mathematics—a role that would make it imperative to find a way to incorporate them into an expanded system of numbers?

欧拉是最早找到这个问题答案的人之一。他是历史上最多产的数学家之一,他拥有惊人的记忆力和非凡的奉献精神,甚至失明也无法阻止他。1735 年,年仅 28 岁的他首先失去了右眼视力,59 岁时几乎失明,但在秘书以及子孙的帮助下,他继续研究,直到 76 岁时突然去世。他曾是令人敬畏的约翰·伯努利(或让·伯努利)的学生——伯努利是莱布尼茨的坚定捍卫者和牛顿的鞭策者——他是第一批用符号微积分明确改写牛顿定律的人之一。他于 1727 年开始与伯努利一起研究复数,但他在这个主题上的工作真正开始形成是在 18 世纪 40 年代,当时他将i与“圆”函数 sin θ 和 cos θ 联系起来。 (请耐心听我说完,因为圆与旋转相关,掌握旋转代数将引导汉密尔顿创建矢量。但首先他需要理解旋转和i之间的联系。)

One of the first to find answers to this question was Euler. He was one of the most prolific mathematicians in history, and he had such a phenomenal memory and such extraordinary dedication that not even blindness stopped him. He lost sight first in his right eye, in 1735 when he was only twenty-eight, and became almost blind at fifty-nine, but with the help of a secretary and his children and grandchildren, he continued researching until he died suddenly at seventy-six. He had been a student of the formidable Johann (or Jean) Bernoulli—the fierce defender of Leibniz and scourge of Newton—and he was one of the first to rewrite Newton’s laws explicitly in terms of symbolic calculus. He began working on complex numbers with Bernoulli in 1727, but his work on this topic really began to come together in the 1740s, when he connected i with the “circular” functions sin θ and cos θ. (Bear with me here, because circles are related to rotations, and coming to grips with the algebra of rotations is what will lead Hamilton to create vectors. But first he will need to understand the connection between rotations and i.)

如果你将 sinθ 和 cosθ 表示在一个半径为 1 的圆上,如图 3.4所示,那么利用毕达哥拉斯定理,你可以得到有用的恒等式

If you represent sinθ and cosθ on a circle with radius 1, as in figure 3.4, then, using Pythagoras’s theorem, you get the helpful identity

(余弦θ) 2+ (罪θ) 2 =1。

(cos θ)2 + (sin θ)2 = 1.

欧拉意识到可以将此方程分解为:

Euler realised that you could factorise this equation as:

(cos θ + i sin θ)(cos θ − i sin θ) = 1。

(cos θ + i sin θ)(cos θ − i sin θ) = 1.

从那里开始,他使用e的一个不成熟的定义(你可以在尾注12中看到),得出了现在被称为欧拉公式的公式:

From there, using a fledgling definition of e (which you can see in the endnote12), he came up with what is now known as Euler’s formula:

e i θ = cos θ + i sin θ。

eiθ = cos θ + i sin θ.

(如果你像欧拉一样知道 sinθ、cos θ 和e θ的泰勒级数,那么这一点就很容易理解了,但这些细节对于我们的故事并不重要。)

(It follows easily if you know, as Euler did, the Taylor series for sinθ, cos θ, and eθ, but these details are not important for our story.)

欧拉的这个非凡公式可以做很多事情。它能让你做纯数学的事情,比如求平方、立方和更高的数复数根在第 1 章中提到过。而且它在实际应用中也非常有用,例如模拟电路和其他波状现象——包括量子力学,如薛定谔方程(它控制着表示移动亚原子粒子(如电子)可能状态的“波函数”)。这是因为从数学上讲,波是周期性的循环,例如当你将 θ 旋转一整圈时,你会从 cosθ 和 sin θ 得到这样的循环(如图3.5所示)。

You can do a lot with Euler’s remarkable formula. It enables you to do pure mathematical things such as finding the square-, cube-, and higher roots of complex numbers mentioned in chapter 1. And it turns out to be incredibly useful in practical applications, too, such as modeling electric circuits and other wave-like phenomena—including in quantum mechanics, as in Schrödinger’s equation (which governs the “wave function” representing the possible state of a moving subatomic particle such as an electron). That’s because mathematically, waves are periodic cycles, such as you get from cosθ and sin θ when you turn θ through a whole circle (as in fig. 3.5).

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图 3.4 . 利用毕达哥拉斯定理以及图中所示的 sin θ 和 cos θ 的定义,可得出 (cos θ) 2 + (sin θ) 2 = 1。

FIGURE 3.4. Using Pythagoras’s theorem and the definitions of sin θ and cos θ shown in the diagram, you get (cos θ)2 + (sin θ)2 = 1.

绕圆旋转也将成为理解实数和虚数之间关系的关键。事实上,由于π在圆的周长和面积公式中起着如此重要的作用,你只需在欧拉公式中选择θ=π,就能看出圆和虚数之间存在某种联系。你会得到一个经常被选为数学中最漂亮的方程:

Rotations around a circle will also turn out to be the key to understanding the relationship between real and imaginary numbers. In fact, since π plays such an important role in the formulae for the circumference and area of a circle, you can already see there’s some sort of connection between circles and imaginary numbers simply by choosing θ = π in Euler’s formula. You get what is often voted the most beautiful equation in mathematics:

eπ + 1 = 0。

eiπ + 1 = 0.

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图 3.5 . y = sin θ的图形草图。

FIGURE 3.5. A sketch of the graph of y = sin θ.

它之所以美丽,是因为它简洁优雅,却又意义深远。它只有 7 个符号,而且它们以一种完全出乎意料的方式相互关联:除了基本运算符号 + 和 = 之外,还有自然对数的底数 e、所有圆心的神秘数字 π、最重要的整数 0 和 1,以及虚数i 谁会想到所有这些不同类型的数字都是相关的?这是一个奇迹般的等式,尽管欧拉从未真正写下来。但由于它是欧拉公式的一个明显进步,并且没有记录表明谁是第一个提出它的人,所以今天它被合理地称为欧拉恒等式。13

It’s beautiful because it is elegantly simple, yet profound. It has just seven symbols, and they are interconnected in an entirely unexpected way: aside from the basic operation signs + and = there is e, the base of the natural logarithms; π, the mysterious number at the heart of all circles; 0 and 1, the most important integers; and the imaginary number i. Who’d have thought that all these different types of number were related? It is a marvel of an equation, although Euler never actually wrote it down. But since it is such an obvious step from Euler’s formula, and since there is no record of who was the first to take it, today it is justifiably called Euler’s identity.13

然而,欧拉仍然对复数的性质感到困惑,而且他也没有在用某种“数轴”表示复数方面取得任何进展。所以,没有人真正知道所有这些数字是如何联系在一起的。

Euler was still confused about the nature of complex numbers, though, and he didn’t make any progress on representing them on some sort of “number line,” either. So, no one really knew how all these numbers were connected.

尽管如此,欧拉还是继续以代数方式使用复数,乍一看可能有点令人惊讶:试图证明费马大定理。或者,我应该说,试图证明它的一个特例——除了 0 和 1 之外,没有整数x、y、z使得x 3 + y 3 = z 3。由于这是一个三次方程,因此可以尝试使用卡尔达诺算法来“解决”它,该算法习惯于求出复数。其思想是,当你发现解决方案,然后证明它与问题固有的假设相矛盾——例如x、y、z必须是整数!这个过程称为矛盾证明,在纯数学中非常有用。欧拉采用了与邦贝利类似的方法,通过探索形式为一个+b3(那是,一个+b3)。

Still, Euler did go on to use complex numbers algebraically, in what might at first seem a surprising way: trying to prove Fermat’s last theorem. Or, should I say, trying to prove a special case of it—that aside from 0 and 1 there are no integers x, y, z such that x3 + y3 = z3. Since this is a cubic equation, it is possible to try to “solve” it with Cardano’s algorithm, which has a habit of turning up complex numbers. The idea is that when you find a solution, you then prove that it contradicts the assumptions inherent in the problem—such as the fact that x, y, z have to be integers! This process is called proof by contradiction, and it is very powerful in pure mathematics. Euler took an approach similar to that of Bombelli, by exploring cubes and cube roots of numbers of the form a+b3 (that is, a+ib3).

费马大定理引起了所有人的兴趣,该定理指出,如果n是大于 2 的整数,则不存在非平凡的x、y、z使得x n + y n = z n。这个定理之所以有趣,是因为找到x 2 + y 2 = z 2的解并不困难——古老的普林普顿 322 石板上有完整的解列表。17 世纪 30 年代,费马曾写下著名的笔记,表示他已经找到了关于他猜想的真正绝妙的普遍证明——但如果他真的找到了,也没人能找到。他确实在n = 4 时给出了证明,而欧拉在n = 3 时给出了证明,尽管晚了一个世纪。14

Everyone was intrigued by Fermat’s last theorem, which says that if n is an integer larger than 2, there are no nontrivial x, y, z such that xn + yn = zn. It was intriguing because there was no problem in finding solutions to x2 + y2 = z2—the ancient Plimpton 322 tablet has a whole list of them. In the 1630s, Fermat had famously scribbled a note indicating that he had found a truly marvelous general proof of his conjecture—but if he did, no one has been able to find it. He did give a proof when n = 4, and Euler’s proof for n = 3 was the next step, albeit a century later.14

在接下来的两个世纪里,许多数学家(包括杰曼)都贡献了特殊情况的证明,直到 20 世纪 90 年代,安德鲁·怀尔斯在新思想和强大计算机能力的帮助下证明了整个证明。但欧拉是证明之路上的潮流引领者,这意味着他使用a + ib形式的代数数也引起了人们的注意,尽管其他人已经做了类似的事情——包括法国人让·勒朗·达朗贝尔,他对欧拉没有承认他感到很恼火。欧拉非常擅长发展他人的新兴思想并将其转化为最终形式,就像他对牛顿定律所做的那样——但他对承认来源相当粗心。也许他阅读范围太广,将思想带入了如此深奥的领域,以至于他完全忘记了自己在哪里读过什么。15

Over yet two more centuries, many mathematicians—including Germain—contributed proofs of special cases, until Andrew Wiles proved the whole thing in the 1990s, with the help of new ideas and massive computer power. But Euler was the trendsetter on the road to a proof, and this meant that his use of algebraic numbers of the form a + ib also got noticed, although others had already done something similar—including Frenchman Jean Le Rond d’Alembert, who was miffed that Euler hadn’t acknowledged him. Euler was terrific at developing others’ fledgling ideas and putting them in their definitive form, the way he did with Newton’s laws—but he was rather careless about acknowledging his sources. Perhaps he read so widely and took ideas into such deeper territory that he simply lost track of where he’d read what.15

欧拉没有在复数上取得进一步进展,但他在这两个项目中取得的成就意义重大——将i与圆函数联系起来,并将a + ib视为代数值,如果分别加减实部和虚部,其行为就像普通数字一样。但半个世纪后,威廉·罗文·汉密尔顿 (William Rowan Hamilton) 却在最后一点上遇到了麻烦,因为他认为这就像把苹果和橘子加在一起一样。16

Euler didn’t go any further with complex numbers, but what he achieved with these two projects was significant—linking i with the circular functions, and treating a + ib as an algebraic quantity that behaved like an ordinary number if you added and subtracted the real and imaginary parts separately. But it was this last bit that William Rowan Hamilton had trouble with, half a century later, for he thought it was like adding apples and oranges.16

当然,自欧拉时代以来已经发生了很多事情。特别是数学家终于找到了一种方法来表示这些奇怪而复杂的图 3.6 中的复平面表示法是一种将复平面表示为某种数轴上的数字的表示法。它是几何的而非代数的,但它非常直观,并且在很大程度上帮助数学家们放心地将这些混合体视为数字。在沃利斯的尝试一个多世纪后,这个解决方案终于应运而生,因为几位数学家独立提出了类似于图 3.6中的现代表示法的想法。其中最著名的是 1799 年的挪威测量员卡斯帕·韦塞尔 (Caspar Wessel),尽管他的开创性努力在接下来的一个世纪里基本上被埋没了;大约在同一时间的高斯,尽管他直到 1831 年才发表论文;阿尔冈 (Argand)(其传记不确定,但可能是出生于瑞士的巴黎书店经理让·罗伯特·阿尔冈 (Jean Robert Argand))于 1806 年提出——无论他是谁,我们通常将图 3.6中的“复平面”称为“阿尔冈平面”以纪念他——以及 1828 年的英国人约翰·沃伦 (John Warren)。

Of course, a lot had happened since Euler’s time. In particular, mathematicians had finally found a way to represent these strange, complex numbers on some sort of number line—or rather, as it turned out, a plane. It was geometric rather than algebraic, but it was brilliantly intuitive, and it went a long way toward helping mathematicians feel secure in treating these hybrid beasts as numbers. More than a century after Wallis’s attempt, it was a solution whose time had come, for several mathematicians independently came up with ideas similar to the modern representation in figure 3.6. The most notable of them are the Norwegian surveyor Caspar Wessel in 1799, although his pioneering effort lay largely buried for the next century; Gauss around the same time, although he didn’t publish till 1831; Argand (whose biography is uncertain but who was probably the Swiss-born Parisian bookstore manager Jean Robert Argand), in 1806— and whoever he was, we often call the “complex plane” in figure 3.6 the “Argand plane” in his honour—and Englishman John Warren in 1828.

最后,我一直在谈论的一些不同的想法开始结合在一起:分量、圆、虚数和矢量。

And finally, some of the disparate ideas I’ve been talking about began to come together: components, circles, imaginary numbers, and a glimpse of vectors.

首先,如果你从图 3.4中取出单位圆的概念,并将其叠加到图 3.6中的复平面上,就会得到图 3.7——一个以实数线和虚数线原点为中心的单位圆。数字 +1 对应于 θ = 0;i对应于θ=π2弧度或 90°;−1 对应于 θ = π 或 180°,−i对应θ=3π2或 270°。(或者你可以顺时针旋转,这样 − i对应于θ=π2或 −90° 等。)但最精彩的部分在于,如果你在脑海中沿着圆从点 1 移动到点i,那么你就旋转了 90°。换句话说,旋转 90° 会将数字 1 变成数字i —就像你写 1 × i = i一样。就好像将实数 1 旋转 90° 相当于将其乘以i一样。现在将i处的点旋转90°:你会得到 −1,就像你说i 2 = −1一样。乘以虚数确实似乎只不过是一个简单的旋转!所以,复平面的奇妙之处在于它展示了虚数如何与实数相关联:通过几何旋转。

First, if you take the idea of the unit circle from figure 3.4 and superimpose it on the complex plane in figure 3.6, you get figure 3.7—a unit circle centred at the origin of the real and imaginary number lines. The number +1 corresponds to θ = 0; i corresponds to θ=π2 radians or 90°; −1 corresponds to θ = π or 180°, and −i corresponds to θ=3π2 or 270°. (Or you can go clockwise so that −i corresponds to θ=π2 or −90°, and so on.) But here’s the fascinating part. If you mentally move along the circle from the point 1 to the point i, you’ve rotated through 90°. In other words, a rotation of 90° takes the number 1 to the number i—just as when you write 1 × i = i. It’s as if the act of rotating the real number 1 through 90° is equivalent to multiplying it by i. Now rotate the point at i by 90°: you get −1, just as you do when you say i2 = −1. Multiplication by an imaginary number does indeed seem to be nothing more than a simple rotation! So, the wonderful thing about the complex plane is that it shows just how imaginary numbers can be related to real ones: via geometric rotations.

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图 3.6。阿尔冈或复平面以实数线为横轴,以虚数线为纵轴。此平面上的点 ( x, y ) 表示复数x + iy。该图显示了特定复数a + ib,表示为点 ( a, b )。

FIGURE 3.6. The Argand or complex plane has the real number line as its horizontal axis and the imaginary number line as its vertical axis. Points (x, y) in this plane represent complex numbers x + iy. The diagram shows the particular complex number a + ib, represented as the point (a, b).

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图 3.7 . 虚数乘法,设想为复平面上的旋转。

FIGURE 3.7. Multiplication by imaginary numbers, envisaged as rotations in the complex plane.

此外,一个由实部和虚部组成的复数,表示为a + ib a + bi,随你喜欢),可以用几何形式表示为这个复平面上的一个( a, b )(如图 3.6所示)。如果你事后再想象从原点到这个点的箭头,你会看到它有一个实部和一个虚部。它显然也有一个方向,以及一个通过毕达哥拉斯定理从分量计算出来的量级。(实际上,复数有一个“模数”而不是量级:复数a + bi的模数定义为一个2+b2,类似于后来出现的矢量的大小。)因此,你可以看到,这种表示复数的几何方法的发明者——阿尔冈、高斯、韦塞尔和沃伦——正在接近矢量的概念,就像牛顿所做的那样。牛顿从物理学的角度来处理这个问题,强调力等物理量的大小和方向,而这些十九世纪早期的数学家则从数论的角度来处理这个问题。汉密尔顿需要明确定义矢量,并将这两种方法结合起来。

What’s more, a complex number, made of real and imaginary parts and expressed as a + ib (or a + bi, whichever you prefer), can be represented geometrically as a point (a, b) in this complex plane (as in fig. 3.6). And if you imagine, with hindsight, an arrow from the origin to this point, you see that it has a real and an imaginary component. It obviously has a direction, too, and a magnitude calculated from the components via Pythagoras’s theorem. (Actually, a complex number has a “modulus” rather than a magnitude: the modulus of the complex number a + bi is defined to be a2+b2, analogous to the magnitude of a vector, which came later.) So, you can see that the inventors of this geometrical way of representing complex numbers—Argand, Gauss, Wessel, and Warren—were closing in on the idea of a vector, just as Newton had done. Newton had approached the problem from the perspective of physics, emphasising the magnitude and direction of physical quantities such as force, while these early nineteenthcentury mathematicians were coming at it from a number theory point of view. It would take Hamilton to define vectors explicitly, and to bring the two approaches together.

汉密尔顿介入

HAMILTON STEPS IN

必须承认,在 19 世纪 30 年代,汉密尔顿还有很长的路要走。最初,他只知道沃伦在复平面上的工作,而不知道高斯或其他人的工作。但沃伦的方法启发了汉密尔顿设想一种思考复数及其几何旋转的方法,以便可以用代数方式处理它们,就像你在解方程时处理普通数字一样——而且不会明显混淆苹果和橘子。

In the 1830s, it must be said that Hamilton still had quite a way to go. Initially, he knew only of Warren’s work on the complex plane, not that of Gauss or the others. But Warren’s approach inspired Hamilton to imagine a way of thinking about complex numbers and their geometrical rotations so that they could be handled algebraically, the way you handled ordinary numbers when you wanted to solve equations—and with no overt mixing of apples and oranges.

他用一种相当聪明的方法解决了这个难题。如果你先取任意两个混合数,然后像对待普通二项式一样对它们进行加法和乘法,但i 2 = −1,你会得到以下结果:

He got around the difficulty in a rather clever way. If you start by taking any two of these hybrid numbers and adding and multiplying them as if they were ordinary binomial expressions, except that i2 = −1, you get this:

a + ib)+(c + id)=(a + c)+ ib + d),

a + ib)(c + id)=(acbd)+ iad + bc)。

(a + ib) + (c + id) = (a + c) + i(b + d),

(a + ib)(c + id) = (acbd) + i(ad + bc).

如今,我们在数学课上经常这样做,欧拉、高斯等人也这样做过,尽管当时这似乎有点敷衍,因为当时还没有人对这种算术感到满意——或者我应该说,对这种数字感到满意。正如高斯告诉天文学家彼得·汉森的那样,“1生动地展现在我的灵魂面前,但用语言来表达却非常困难,因为语言只能给出一个悬浮在空中的图像。”汉密尔顿的朋友奥古斯都·德·摩根声称“展示了这个符号1毫无意义,或者说自相矛盾、荒谬可笑。” 17

We do it all the time in maths classes today, and Euler, Gauss, and others had done it, too, although back then it still seemed somehow like fudging, because no one was yet comfortable with this kind of arithmetic—or, should I say, with this kind of number. As Gauss told the astronomer Peter Hansen, “The true meaning of 1 reveals itself vividly before my soul, but it will be very difficult to express it in words, which can give only an image suspended in air.” And Hamilton’s friend Augustus De Morgan claimed to have “shown the symbol 1 to be void of meaning, or rather self-contradictory and absurd.”17

因此,汉密尔顿说,为什么不定义复数时去掉分散注意力的i,即苹果中散落的橘子,同时仍然保留相同的加法和乘法规则呢?为什么不把复数完全看作实数的“对”(或现代数学家所说的“有序对”)—— a ,b),(c,d)等等——然后定义加法和乘法,使它们遵循相同的模式:

So, Hamilton said, why not define complex numbers without the distracting i, the stray orange among the apples, yet still keep the same rules of addition and multiplication. Why not think of complex numbers entirely in terms of “couples” (or what modern mathematicians call “ordered pairs”) of real numbers—(a, b), (c, d), and so on—and then define addition and multiplication so that they followed the same pattern:

a,b)+(c,d)=(a + c,b + d),

a,b)×(c,d)=(acbd,ad + bc)。

(a, b) + (c, d) = (a + c, b + d),

(a, b) × (c, d) = (acbd, ad + bc).

早期数学家,尤其是高斯,已经接近这个想法,但汉密尔顿将其阐述得很清楚。起初并没有取得多大成功。几千年来,乘法只是一个数字乘以另一个数字,所以汉密尔顿的乘法,混合了乘法、加法和减法,是非常不同的东西。

Earlier mathematicians, especially Gauss, had come close to this idea, but Hamilton spelled it out clearly. Not that it met with much success at first. For thousands of years multiplication had been just one number multiplied by another, so Hamilton’s multiplication of couples, with its jumbled-up mixture of multiplications, additions, and subtractions, was something very different.

但时机已到,因为他的英国同事德·摩根和乔治·皮科克已经开始探索完全符号代数的概念——从哈里奥特和沃利斯转向代数的概念,这种代数不仅脱离几何,也脱离算术。换句话说,代数用符号来定义,这些符号不必有任何具体的含义,但必须遵循明确既定的规则——就像汉密尔顿的“配对”一样。皮科克和德·摩根首先正式表达了普通代数的规则,尽管这些规则已经被直观地使用了数百年甚至数千年。毕竟,三个苹果加两个还是两个苹果加三个都无关紧要,所以你会认为它几乎不需要称为“加法交换律”。当然,除非你想把代数从数值转移到纯粹抽象符号和运算的科学,就像皮科克和他的前学生德·摩根所做的那样。

But its time was near, for his English colleagues De Morgan and George Peacock had already begun exploring the idea of a completely symbolic algebra—moving on from Harriot and Wallis to an idea of algebra divorced not just from geometry but from arithmetic, too. In other words, algebra defined in terms of symbols that didn’t have to mean anything concrete, but which did have to obey clearly established rules—just as Hamilton’s “couples” do. Peacock and De Morgan had begun by expressing the rules of ordinary algebra formally, even though they’d been used intuitively for hundreds, perhaps thousands of years. After all, it doesn’t matter if you add three apples to two or two apples to three, so you’d think it hardly needed stating as the “commutative law for addition.” Unless, of course, you wanted to move algebra beyond numerical quantities to a science of purely abstract symbols and operations, as Peacock and his former student De Morgan did.

皮科克对符号系统的兴趣最初是在微积分的背景下形成的。在第二章中,我提到了与牛顿的点符号相比,莱布尼茨微分dx、dy等富有启发性的威力,而皮科克曾是一群年轻的剑桥学者中的一员,其中包括因原型计算机而出名的查尔斯·巴贝奇和约翰·赫歇尔,后者是发现天王星的著名天文学家的儿子,也是一位著名的天文学家。他们鼓吹课程改革,用莱布尼茨的微积分取代牛顿的微积分。正如巴贝奇顽皮地说的那样——既是对正式宗教的嘲讽,也是对剑桥大学对其昔日学生牛顿的敬畏——这些特立独行的年轻人支持“纯粹的d主义原则,而不是大学的点时代” 。18

Peacock’s interest in symbolism had initially taken shape in the context of calculus. In chapter 2, I mentioned the power of the suggestive Leibnizian differentials dx, dy, and so on, compared with Newton’s dot notation, and Peacock had been part of a group of young Cambridge academics—including Charles Babbage of prototype-computer fame, and John Herschel, son of the famous astronomer who discovered Uranus, and a significant astronomer in his own right—who’d agitated for a curriculum reform to replace Newton’s calculus with Leibniz’s. As Babbage impishly put it—punning on both formal religion and the awe in which Cambridge still held its former student Newton—these young mavericks supported “the principles of pure d-ism as opposed to the dot-age of the university.”18

至于纯符号代数,汉密尔顿本人最初横跨两个阵营。他同意复平面上的几何表示是说明性的,而不是从根本上讲是数学的。但他并不是一个纯粹的符号主义者,因为他认为代数应该是比一套规则更具体的东西。他甚至想知道这两个阵营是否像空间和时间一样相互联系。德国哲学家伊曼纽尔康德曾说过,我们的存在取决于预先存在的空间和时间,几何学是空间表示的基本语言——所以,汉密尔顿想,也许代数是时间的科学?毕竟,他说,我们标记时间和空间的方式是“思想变成事物,精神穿上身体,心灵的行为和激情被赋予外在存在,我们从远处观察自己”的手段。19

As for purely symbolic algebra, Hamilton himself initially straddled both camps. He agreed that geometrical representations on the complex plane were illustrative rather than fundamentally mathematical. But he wasn’t a pure symbolist, because he felt that algebra should be about something more tangible than a set of rules. He’d even wondered if the two camps were linked like space and time. The German philosopher Immanuel Kant had said that our existence depends on preexisting space and time, and that geometry is the fundamental language of spatial representation—so, thought Hamilton, perhaps algebra was the science of time? After all, he said, the way we mark time and space is the means “by which thoughts become things, and spirit puts on body, and the act and passion of mind are clothed with an outward existence, and we behold ourselves from afar.”19

抛开诗歌不谈——汉密尔顿从年轻时就开始写诗——如今我们已经习惯了用代数方法处理任意的xyab或我们选择的任何符号,他的时间代数听起来很奇怪。他的符号主义同事也不喜欢它。但正如莱布尼茨和其他人很久以前指出的那样,我们的时间概念(以及空间概念)是相互关联的——这件事发生在那件事之前,这件事正在发生等等。爱因斯坦将这个想法推向了合乎逻辑的结论,在这个过程中提出了一些关于时间变慢和“现在”本身是相对的奇怪新想法;与此同时,汉密尔顿意识到这些关系时间时刻之间的时间差可以用代数表达式来表达。他的重点是纯数学而不是物理学——否则他可能已经预见到了爱因斯坦;事实上,他引用了牛顿的话,牛顿在描述事物如何变化时,曾诉诸于增加的“时刻”和时间的“流动”,他提出了我们现在称之为导数的概念,但他当时称之为“流动数”。汉密尔顿将时间类比带入了更基本的领域,他建议,如果你把数字本身看作表示有向线段上“时刻”之间的心理“步骤”,你就可以“解释”算术定律作为一门时间科学。

Poetry aside—Hamilton had been writing poetry since his youth—his algebra of time sounds odd today, when we are so used to algebraically manipulating arbitrary x’s and y’s or a’s and b’s or whatever symbols we choose. His symbolist colleagues didn’t like it either. But as Leibniz and others had pointed out long ago, our notion of time (and of space, too) is relational—this happened before that, this is happening now, and so on. Einstein would take this idea to its logical conclusion, turning up in the process some bizarre new ideas about time slowing down and “now” itself being relative; in the meantime, Hamilton realised that these relations between moments of time could be expressed in terms of algebraic expressions. His focus was pure mathematics rather than physics—otherwise he might have anticipated Einstein; as it is, he cited Newton, who had appealed to “moments” of increase and the “flux” of time when describing how things change, in his concept of what we now call a derivative but which he had called a “fluxion.” Hamilton took the time analogy into more basic territory by suggesting that if you think of numbers themselves as representing the mental “steps” between “moments” on a directed line segment, you can “explain” the laws of arithmetic as a science of time.

这些“步骤”相当深奥,尽管后来他从时间中的瞬间转移到空间中的点,并将他的“步骤”定义为向量;无论如何,你可以通过一个例子来了解他的意思:令人困惑但似乎不可避免的事实是两个负数的乘积是正数,因此 (-3) × (-1) 等于 3。正如汉密尔顿所说,在他的哲学中,你可以用“两次连续的逆转恢复了步骤的方向”来解释这个奇怪的结果。这里的要点是方向在他的矢量概念中的重要性:将箭头的方向反转两次,你就会回到起点。这类似于说“我不会不去”——两个连续的“不”相互抵消,就像两个减号取消或“恢复”我们到正方向一样。当你将两个小于零的数字相乘但最终得到正数时,这当然有助于理解事情!20

These “steps” were rather esoteric, although later he moved on from moments in time to points in space, and he identified his “steps” as vectors; either way, you can get his drift in an example: the confusing but seemingly inescapable fact that the product of two negative numbers is positive, so that (−3) × (−1), say, is equal to 3. As Hamilton put it, in his philosophy you could explain this bizarre result by saying that “two successive reversals restore the direction of a step.” The take-home message here is the importance of direction in his conception of a vector: reverse the direction of an arrow twice and you’re back where you started. It’s analogous to saying “I will not not go”—two successive “nots” cancel each other out, just as two minus signs cancel or “restore” us to the positive direction. It certainly helps make sense of things when you’re multiplying two numbers that are each less than zero and yet you end up with a positive!20

汉密尔顿最终确实更接近于纯符号方法,今天看来,这在他对“耦合”算术的定义中显而易见。德摩根也几乎明白这一点。他“倾向于认为”,如果你抛开时间,只关注实数,汉密尔顿“可能最终”提供了一种通过符号代数而不是直观的几何类比来思考复数的方法。21

Hamilton did eventually move closer to the purely symbolic approach that seems evident, today, in his definition of the arithmetic of “couples.” It had been almost evident to De Morgan, too. He was “inclined to think” that if you left time out of it and focussed simply on real numbers, Hamilton “may finally” have provided a way to think about complex numbers through symbolic algebra, not just intuitive geometric analogy.21

如今,我们很容易想知道为什么有人会担心负数和复数的特殊性以及代数规则——我们已经习惯了在需要的时候使用这些东西,而不用担心潜在的概念问题。但当然,这是因为一些最伟大的数学家确实担心正是这些事物,我们才有了现代数学和许多技术。汉密尔顿在 1837 年发表的关于复杂耦合的论文中,一开始就说他的重点是纯粹的数学。他当时并不知道,今天 NASA 会利用他的研究成果来帮助驱动其航天器!所以,即使他最初对时间的关注似乎是错误的,但正是这种打破常规的思维——对数字和算术基础的关注——最终让汉密尔顿发明了向量和四元数,而它们在今天有着如此多的实际应用。

Today, it’s easy to wonder why anyone would worry about the peculiarities of negative and complex numbers and the rules of algebra—we are so used to just getting on with it and using these things when we need to, without worrying about underlying conceptual problems. But, of course, it is because some of the greatest mathematical minds did worry about such things that we have both modern maths and much of our technology. When Hamilton introduced his paper on complex couples, which he published in 1837, he said right up front that his focus here was purely mathematical. Little did he know that today NASA would be using the ultimate fruits of his research to help drive its spacecraft! So even if his initial focus on time seems misguided, it was this kind of out-of-the-box thinking—this concern with the very foundations of number and arithmetic—that would ultimately lead Hamilton to the invention of vectors and quaternions, which have so many practical applications today.

值得注意的是,汉密尔顿的灵感来源不仅包括数学家,还包括康德等哲学家,以及作家和诗人:威廉·华兹华斯是他的好友,著名小说家和教育理论家玛丽亚·埃奇沃思也是他的好友,浪漫主义者弗朗西斯·博福特·埃奇沃思(玛丽亚同父异母的弟弟)和塞缪尔·泰勒·柯勒律治也是他的好友。华兹华斯说,汉密尔顿和柯勒律治是他见过的最出色、最有天赋的两个人。玛丽亚·埃奇沃思第一次见到当时只有 19 岁的汉密尔顿时,她认为他“既有真正的天才的朴素,又有真正的天才的坦率” 。22

It is telling that Hamilton’s sources of inspiration included not just mathematicians but also philosophers such as Kant—and writers and poets, too: William Wordsworth was a good friend, and so were the famous novelist and educational theorist Maria Edgeworth along with the Romantics Francis Beaufort Edgeworth (Maria’s younger half-brother) and Samuel Taylor Coleridge. Wordsworth said Hamilton and Coleridge were the two most wonderful and gifted men he had ever met. And when Maria Edgeworth first met Hamilton, who was then only nineteen, she thought he had “both the simplicity and the candour which make a true genius.”22

这种兼收并蓄的影响在十九世纪并不像今天这样罕见,因为如今数学和科学一样,已经变得如此复杂和专业化。但即使在那个年代,汉密尔顿也不是一个普通人。

Such eclectic influences were not as unusual in the nineteenth century as they are today, now that mathematics, like science, has become so sophisticated and specialised. But Hamilton was no ordinary man, even in those days.

耀眼的天才

A DAZZLING PRODIGY

1805 年 8 月 3 日午夜,威廉出生在都柏林。父亲阿奇博尔德是一名律师,母亲莎拉则被认为才华横溢。威廉是五个孩子中唯一的儿子,天赋异禀,因此父母将威廉托付给了受过大学教育的叔叔詹姆斯·汉密尔顿,他是一位圣公会牧师、古典学家、语言学家和教区校长。从三岁起,威廉大部分时间都和叔叔住在一起,叔叔对他早熟的才华培养得非常好,据说到他十三岁时,他就已经精通希腊语、拉丁语、希伯来语、法语、德语和意大利语,并且正在学习梵语、波斯语和阿拉伯语。

He was born in Dublin at midnight, as August 3 turned into August 4, 1805. His father, Archibald, was an attorney, and his mother, Sarah, was considered intellectually gifted. But William—the only son among five surviving children—was so gifted that his parents entrusted his education to his university-educated uncle, James Hamilton, an Anglican clergyman, classicist, linguist, and diocesan schoolmaster. From the age of three, William lived mostly with his uncle, who nurtured his precocious talent so well that by the time he was thirteen, he reputedly had a good knowledge of Greek, Latin, Hebrew, French, German, and Italian, and was studying Sanskrit, Persian, and Arabic.

虽然詹姆斯叔叔对威廉同样非凡的计算天赋并不十分在行,但他还是尽力为威廉寻找合适的教科书。年轻的威廉因此能够利用他的法语技能,通过法语教科书自学微积分——所以他学习的是莱布尼茨微积分,而不是牛顿形式——并且在十六岁时,他开始学习皮埃尔-西蒙·拉普拉斯的《天体力学》。这是牛顿《自然哲学的数学原理》的一次重大更新,吸收了过去一个世纪数学天文学的所有进展。拉普拉斯是一位伟大的数学家,但十几岁的汉密尔顿发现他推理中的一个缺陷——恰好是关于合成力的平行四边形规则的。显然,他已经在开拓矢量的道路上前进了。

Uncle James wasn’t quite so able when it came to William’s equally prodigious talent for calculating, but he did his best to find suitable textbooks for him. Young William was thereby able to use his French skills to teach himself calculus from a French textbook—so he learned Leibnizian calculus rather than Newton’s form—and at sixteen he began to study Pierre-Simon Laplace’s Traité de Mécanique Celeste (Treatise on Celestial Mechanics). This was a monumental update of Newton’s Principia, incorporating all the advances in mathematical astronomy that had taken place in the intervening century. Laplace was a great mathematician, but the teenaged Hamilton spotted a flaw in his reasoning—about, as it happens, the parallelogram rule for composing forces. Evidently, he was already on the road to pioneering vectors.

那是 1821 年。与此同时,一位自学成才的苏格兰女性也在私下研究《天体力学》,十年后,她以教科书《天体力学》震惊英国数学界。她就是玛丽·萨默维尔——我们之前见过她,当时她九十多岁,正在研究汉密尔顿的四元数——她的书是拉普拉斯五卷本前两卷的扩展和详尽的 610 页英文版,外加一份 70 页的“初步论文”,为非专业人士提供概述。玛丽亚·埃奇沃思读了这篇概述后被迷住了:她发现她朋友萨默维尔的写作风格简单,非常适合“科学的崇高”。至于学术界,乔治·皮科克是剑桥大学数学家之一,他对整本书印象深刻,以至于他们用它作为高级天文学学生的教科书。 (诚​​然,书中对莱布尼茨微积分的使用也推动了皮科克的课程改革——但这本书太优秀了,以至于在下个世纪一直是标准教材。)萨默维尔非常激动——尤其是当评论家评论说,一位自学成才的女性写了一本很少有男性能理解的书,这是多么了不起。但她也非常愤怒,因为女性仍然被剥夺了接受良好教育的权利——在 19 世纪 60 年代末,她是第一个签署约翰·斯图尔特·密尔(最终未获成功)的女性投票权请愿书的人。23

This was in 1821. At the same time, a self-taught Scottish woman was also privately studying Mécanique Celeste, and ten years later she would astonish the British mathematical community with her textbook Mechanism of the Heavens. She was Mary Somerville—we met her earlier, when she was studying Hamilton’s quaternions in her nineties—and her book was an expanded, explicated 610-page English version of the first two volumes of Laplace’s five-volume tome, plus a 70-page “preliminary dissertation” giving an overview for the nonspecialist. Maria Edgeworth read the overview and was entranced: she found that the simplicity of her friend Somerville’s writing superbly suited “the scientific sublime.” As for the academics, George Peacock was one of the Cambridge mathematicians who were impressed with the whole book—so much so that they used it as a text for their advanced astronomy students. (True, its use of Leibnizian calculus also helped further Peacock’s curriculum reform—but the book was so good that it remained the standard text for the next century.) Somerville was thrilled—especially when reviewers commented on how extraordinary it was that a self-taught woman had written a book that few men could understand. But she was furious, too, because women were still denied a decent education—and in the late 1860s she would be the first to sign John Stuart Mill’s (ultimately unsuccessful) petition for votes for women.23

相比之下,汉密尔顿确实上了大学——就读于都柏林圣三一学院,那里的古典文学和数学教学都非常出色。他的表现非常出色,不到 22 岁就被任命为爱尔兰天文学教授和皇家天文学家。几年后,正如我所提到的,他因对圆锥折射的数学预测而闻名。然后,还在 20 多岁时,他就提出了一种新的动力学解释,这种解释在物理学的许多领域都具有重要意义——例如,在运动物体系统的力学中,以及在描述电子 (量子) 力学的薛定谔方程中。1833 年,当他向爱尔兰皇家科学院提交关于复数“对”的初步想法时,他才 28 岁,30 岁时,他因对数学的贡献而被授予爵士称号。

Hamilton, by contrast, did go to university—to Trinity College Dublin, where the teaching was excellent, in both classics and mathematics. He did so brilliantly that he was made professor of astronomy and royal astronomer of Ireland when he was not quite twenty-two. A few years later he became famous for his mathematical prediction of conical refraction, as I’ve mentioned. Then, still in his twenties, he formulated a new interpretation of dynamics that would become significant in many areas of physics—in the mechanics of systems of moving objects, for instance, and in Schrödinger’s equation describing the (quantum) mechanics of an electron. He was still only twenty-eight when he presented his initial ideas on complex number “couples” to the Royal Irish Academy in 1833, and he was knighted at thirty for his services to mathematics.

1837 年底,32 岁的汉密尔顿当选为爱尔兰皇家学院院长,他做的第一件事就是向刚满 70 岁的玛丽亚·埃奇沃思寻求建议。学院旨在培养科学和人文学科——埃奇沃思的父亲是学院的创始人之一——但汉密尔顿认为文学没有得到足够的重视。“全世界都知道,”他写信给她,“你不仅是一个文学爱好者,还是一个成功的文学追求者和强有力的推动者,在与文学相关的任何方面,你的意见一定是最有价值的。”埃奇沃思的作品今天可能已被人们遗忘——尽管其中两部作品在 2009 年被《卫报》列为“每个人都必须阅读的 1000 部小说”——但在 19 世纪初,她是爱尔兰最著名的小说家,不仅汉密尔顿崇拜她,沃尔特·斯科特和简·奥斯汀等人也崇拜她。她给了汉密尔顿一些中肯的建议——比如为征文比赛提供奖牌,将订阅费定在普通文学爱好者可以承受的水平——汉密尔顿在就职演讲中感激地引用了其中的大部分建议。不幸的是,他没有足够的勇气去接受她关于允许女性参加学院特别讨论会的建议。她长期以来一直在倡导女性受教育的权利和参与科学的权利,而汉密尔顿在回复中找的借口让她很不满意。尽管如此,他还是主持了她 1842 年的学院荣誉院士选举——她是第一位获得此殊荣的本地女性。此前已有三名外国女性获得这一殊荣——包括玛丽·萨默维尔和卡罗琳·赫歇尔(约翰·赫歇尔的姨妈,也是和她一样的天文学家)她们是英国女童军的一名成员(包括她的侄子和她的兄弟威廉)——但由于她们生活在国外,她们不太可能因为想要参加会议而惹恼男性。直到 1949 年,第一批正式女性成员才被接纳。24

At the end of 1837, thirty-two-year-old Hamilton was elected president of the Royal Irish Academy, and one of the first things he did was seek advice from Maria Edgeworth, who had just turned seventy. The Academy was designed to foster science and the humanities—Edgeworth’s father had been a founder—but Hamilton felt that literature wasn’t getting enough of a look-in. “It is known to all the world,” he wrote to her, “that you are not only a lover of literature but a successful pursuer and powerful promoter of it, and that on any point connected therewith, your opinion must be most valuable.” Edgeworth’s works may be largely forgotten today—although in 2009 two of them made the Guardian’s list of “1000 novels everyone must read”—but in the early nineteenth century she was Ireland’s most famous novelist, and it wasn’t just Hamilton who admired her but the likes of Walter Scott and Jane Austen, too. She gave Hamilton some sound advice—such as offering medals for essay competitions, and setting subscription fees at an affordable rate for ordinary literaturelovers—and he gratefully drew on most of it in his inaugural presidential address. Unfortunately, he wasn’t brave enough to tackle her advice to allow women to attend special Academy discussion nights. She had long been promoting women’s right to education and their right to engage with science, and she wasn’t impressed with the excuses he made in his reply to her. Still, he would preside over her election as an honorary member of the Academy in 1842—the first local woman to be so honoured. Three foreign women had already achieved this distinction—including Mary Somerville and Caroline Herschel ( John Herschel’s aunt, and an astronomer like her nephew and her brother William)—but, living abroad, they were unlikely to ruffle male feathers by wanting to attend meetings. The first full female members would not be admitted until 1949.24

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图 3.8A . 用旋转表示的复数乘法。

FIGURE 3.8A. Multiplication of complex numbers represented by rotations.

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图 3.8B。如何通过旋转计算乘法:使用极坐标和欧拉公式,写出x + iy = r cos θ + ir sin θ = rei θ,以及a + ib = s cos α + is sin α = sei α;然后,利用指标定律,可知乘积为rsei (θ+α)。因此,x + iy乘以a + ib扩展了原始向量的幅度,并将其旋转了 α 弧度。简单地将一个复数添加到另一个复数上可以看作是平面上的平移。

FIGURE 3.8B. How to calculate multiplications via rotations: Use polar coordinates and Euler’s formula to write x + iy = r cos θ + ir sin θ = reiθ, and a + ib = s cos α + is sin α = seiα; then, using the index laws, you see that the product is rsei(θ+α). So, multiplication of x + iy by a + ib has extended the magnitude of the original vector and rotated it through an angle of α radians. Simply adding one complex number to another can be viewed as a translation in the plane.

令人失望的是,汉密尔顿在所谓的女性问题上并不像在数学上那样领先于时代,但他接下来的所作所为对于向量的故事至关重要。首先,虽然高斯和其他人已经看到复数可以表示为复平面上的一个,但汉密尔顿将其视为“有向线段”——换句话说,一个箭头。25韦塞尔也有类似的想法,但汉密尔顿和主流中的其他人都不知道。阿尔冈德也考虑过“有向线段”,但汉密尔顿的下一步才是向量的真正基础。如果你将两个“对”——阿尔冈德平面上的两个“箭头”(x,y)和(a,b)相乘——你会得到二维平面上的旋转,如图 3.8所示。所以复数乘法是与旋转的几何运算有关——就像乘以i一样。汉密尔顿想知道,是否可以通过“三元组”的乘法来表示三维空间中的旋转?

Disappointing as it is that Hamilton was not as ahead of his time on the so-called woman question as he was in mathematics, it is what he did next that is important for the story of vectors. First, while Gauss and others had seen that a complex number could be represented as a point in the complex plane, Hamilton saw it as a “directed line segment”—in other words, an arrow.25 Wessel had something like this idea, too, but neither Hamilton nor anyone else in the mainstream knew about it. Argand also considered “directed line segments,” but it was Hamilton’s next step that would be truly foundational for vectors. If you multiply two “couples”—two “arrows” (x, y) and (a, b) in the Argand plane—you get a rotation in a twodimensional plane, as in figure 3.8. So complex number multiplication is linked to the geometrical operation of rotation—just as it is for multiplication by i. Was it possible, Hamilton wondered, to represent rotations in three-dimensional space via the multiplication of “triples”?

1841 年,他开始认真研究这个问题,当时他读到一篇论文,其中他的朋友德·摩根 (De Morgan) 总结说,用代数表示三维几何似乎根本不可能。虚数i在二维空间中发挥了旋转的魔力,但德·摩根认为,没有更多这样的符号可用——没有新的代数思想可以将几何从二维页面转移到整个空间。

He began working on the problem in earnest in 1841, when he read a paper in which his friend De Morgan concluded that it simply didn’t seem possible to represent three-dimensional geometry algebraically. The imaginary number i had performed its rotational magic in two dimensions, but De Morgan argued that there were no more such symbols available—no new algebraic ideas that could move geometry from the two-dimensional page to all of space.

这正是汉密尔顿所需要的挑战。

It was just the challenge Hamilton needed.

(4)了解空间(和存储)

(4) UNDERSTANDING SPACE (AND STORAGE)

汉密尔顿多年来一直在思考他的三元组代数,到 1843 年秋天,甚至他的孩子们也开始关注他的传奇故事。尽管他们还很小,汉密尔顿还是试图向八岁的阿奇博尔德和九岁的威廉·埃德温解释他的基本思想,现在每天早上吃早餐时,他们都会问:“爸爸,你会三元组吗?”汉密尔顿每天早上都会悲伤地摇头说:“不,我只会加减。” 1

Hamilton mulled over his algebra of triples for years, and by the fall of 1843 even his children were following the saga. Although they were very young, Hamilton had tried to explain his basic idea to eight-year-old Archibald and nine-year-old William Edwin, and now every morning at breakfast they would ask, “Well, Papa, can you multiply triplets?” And every morning Hamilton would shake his head sadly, saying, “No, I can only add and subtract them.”1

汉密尔顿的几封信都动人地提到了他向“我的孩子们”解释数学的故事,就三元组而言起初,答案似乎是小菜一碟——至少对汉密尔顿这样的神童来说是这样。毕竟,乘以一对复数实数(x,y)相当于在二维平面上旋转有向线OP = x + iy ,如图3.6 - 3.8所示——所以在三维空间中你肯定会有“三元组”(x,y,z),而你只需旋转OP = x + iy + jz。当然,你必须发明j,另一个像i一样的虚数,但这似乎很简单:如果你将阿尔冈平面扩展到三维,j在第三轴方向上,你会得到一个新的虚数,由于方向不同,它与i不同。如果你通过类比i来定义j的乘法——即在x - z平面上旋转——你会得到j 2 = −1。(事实上,今天的工程师用j来表示1,因为他们用i来表示电流。)但是,这个新数字j与原来的i之间的代数差是什么?

Several of Hamilton’s letters have touching references to his explaining maths to “my boys,” and in the case of triples it had seemed, at first, as if the answer would be child’s play—for a prodigy such as Hamilton, at least. After all, multiplying by a complex “couple” of real numbers (x, y) was equivalent to rotating the directed line OP = x + iy in the two-dimensional plane, as in figures 3.63.8—so surely in 3-D space you’d have “triples” (x, y, z), and you’d just be rotating OP = x + iy + jz. You’d have to invent j, of course, another imaginary number like i, but that seemed straightforward enough: if you extended the Argand plane to three dimensions, with j in the direction of the third axis, you’d have a new imaginary number, different from i because of its direction. And if you defined multiplication by j via analogy with i—that is, as a rotation in the x-z plane—you’d have j2 = −1. (In fact, today engineers use j to denote 1, because they use i to denote electric current.) But then, what was the algebraic difference between this new number j and the original i?

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威廉·罗文·汉密尔顿爵士和他的一个儿子(约 1845 年);经爱尔兰皇家学院许可 © RIA。当时,拍照需要长时间曝光,这可能解释了为什么汉密尔顿从不微笑;但据说,他很有幽默感。

Sir William Rowan Hamilton with one of his sons (circa 1845); by permission of the Royal Irish Academy © RIA. In those days photographs required long exposure times, which likely explains why Hamilton is never smiling; but by all accounts, he had a great sense of humour.

汉密尔顿做的第一件事就是研究代数定律,德·摩根和皮科克已经开始将其形式化。这似乎是显而易见的。如果你将数字相加,如x + iy + jza + ib + jc,那么以何种顺序相加并不重要,而且你可以相当肯定结合律也会成立。(我在尾注中对这些定律进行了提醒。2

The first thing Hamilton did was to check in with the laws of algebra, which De Morgan and Peacock had begun to formalise. It seemed obvious that if you added numbers like x + iy + jz and a + ib + jc, it wouldn’t matter in which order you added them, and you could be pretty sure that the associative law would hold, too. (I’ve given a reminder of these laws in the endnote.2)

然后是代数“闭包”的概念,正如现代数学家所说的那样。如果将两个整数相加,会得到另一个整数;如果将两个实数相加,会得到另一个实数;如果将两个复数相加,会得到另一个复数。很明显,如果将两个“三元组”相加,闭包也成立。所以这一切都很简单。乘法会很棘手。

Then there’s the idea of algebraic “closure,” as modern mathematicians put it. If you add two integers, you get another integer; if you add two real numbers, you get another real number; and if you add two complex numbers, you get another complex number. It’s obvious that closure holds if you add two “triples,” too. So that was all nice and easy. It was multiplication that was going to be tricky.

汉密尔顿通过展开( x + iy)(a + ib )找到了乘以他的数对的规则。但是,如果逐项相乘(x + iy + jz)和(a + ib + jc),您会发现不仅需要i 2 = j 2 = −1,还必须弄清楚ijji的含义。从表面上看,这些额外的项表明三元组的乘法不是封闭的 - 对于纯数学家来说,这意味着三元组不形成一致的乘法代数。(如果您永远不知道乘积中会得到什么类型的数字,那么您就无法设计出适用于初始集合中每个可能数字的一致代数规则 - 如果没有规则,您就无法解方程。)

Hamilton had found the rules for multiplying his couples simply by expanding (x + iy)(a + ib). But if you multiply out (x + iy + jz) and (a + ib + jc), term by term, you can see that you’re not only going to need i2 = j2 = −1, you’re also going to have to figure out what ij and ji mean. On the face of it, these extra terms suggest that multiplication of triples is not closed— and to a pure mathematician, that means triples do not form a consistent multiplicative algebra. (If you never know what type of number you’ll get in a product, you can’t devise consistent algebraic rules that work for every possible number in your initial set—and if there are no rules, you can’t solve equations.)

按照良好的数学研究风格,汉密尔顿首先试图简化问题,假设ij = 0。这当然会使乘法封闭,但这有点牵强——在普通算术和代数中,不能将两个非零量相乘得到零。(如果一个或两个非零矩阵的行列式为零,则可以从两个非零矩阵的乘积中得到零,例如当矩阵方程AX = 0 时——但矩阵代数是在四元数之后出现的。)还有其他定律需要检查——汉密尔顿特别担心他所谓的“模数定律”,我在尾注中对此进行了解释。3然而,为了让这个定律适用于三元组,汉密尔顿发现ij不能等于0。正如他告诉约翰·格雷夫斯——他的大学老友,他也一直在考虑扩展复平面,他的兄弟罗伯特后来写了一本对汉密尔顿充满同情的传记——他随后绝望地尝试了ij = − ji,尽管它与普通算术相反,就像ij = 0 一样。他甚至尝试了ij = jji = i。但是,尽管所有这些奇怪的可能性确实简化了乘法的某些方面,但他的新数字仍然不满足模数定律。4

In good mathematical research style, Hamilton started out by trying to simplify matters, assuming ij = 0. That certainly would make multiplication closed, but it was a stretch—in ordinary arithmetic and algebra, you can’t multiply two nonzero quantities and get zero. (You can get zero from the product of two nonzero matrices if the determinant of one or both is zero, such as when you have the matrix equation AX = 0—but matrix algebra came after quaternions.) And there were other laws to check, too— Hamilton was worried especially about what he called the “law of moduli,” which I’ve explained in the endnote.3 To make this work for triples, however, Hamilton found that ij could not equal 0. As he told John Graves—his old friend from university, who had also been thinking about extending the complex plane, and whose brother Robert would later write a sympathetic biography of Hamilton—he then tried ij = −ji in desperation, even though it was as contrary to ordinary arithmetic as ij = 0. He even tried ij = j and ji = i. But while all these bizarre possibilities did simplify some aspects of multiplication, his new numbers still didn’t satisfy the law of moduli.4

汉密尔顿一定很想放弃。也许这真的是不可能的,就像他的朋友德·摩根所说的那样。

Hamilton must have been sorely tempted to give up. Perhaps it really was impossible, just as his friend De Morgan had said.

一段数学友谊、两次婚姻和一次最终实现的探索

A MATHEMATICAL FRIENDSHIP, TWO MARRIAGES, AND A QUEST FINALLY FULFILLED

德·摩根是个有趣的人,除了他在抽象代数基础上的开创性工作之外。他生性活泼,思想开明,没有正式的宗教信仰——即使在学校,他也更喜欢在教堂的长椅上计算方程式,而不是听布道。他认为自己是一个“不执着的”基督徒,热衷于宗教和思想自由——尽管他曾在牛顿的母校三一学院学习,但他拒绝宣誓,而这是剑桥和牛津大学奖学金的必要条件。牛顿同样拒绝宣誓,不是因为他支持宗教宽容——远非如此——而是因为他是秘密的阿里乌斯派,这意味着他不相信耶稣是神。但他非常聪明,国王给了他特别豁免,他还是获得了剑桥奖学金。德·摩根很快就会发表一些关于牛顿的精辟文章。他非常钦佩牛顿的天才和他的基本道德。但他的正义感也迫使他直言这位伟大数学家的缺点,尤其是与莱布尼茨的优先权之争。这是很少有人敢做的事,因为牛顿的头脑看起来如此神圣,两个世纪以来他一直受到如此崇拜。5

De Morgan was an interesting person, quite aside from his pioneering work on the foundations of abstract algebra. Vivacious and liberal-minded, he hadn’t taken to formal religion—even at school he’d preferred pricking out equations on the pews to listening to the sermons. He considered himself an “unattached” Christian and was passionate about religious and intellectual freedom—so much so that although he’d studied at Newton’s old college, Trinity, he refused to take the Anglican oaths that were required for fellowships at Cambridge and Oxford. Newton had likewise refused the oath, not because he was in favour of religious tolerance—far from it—but because he was a secret Arian, which meant he didn’t believe in the divinity of Jesus. But he was so brilliant that the king had given him a special dispensation and he got his Cambridge fellowship anyway. De Morgan would soon publish some astute essays on Newton. He admired immensely both Newton’s genius and his essential morality. But his own sense of justice compelled him to speak out about the great mathematician’s flaws, too, particularly in connection with the priority dispute with Leibniz. This was something very few had dared to do, so godlike did Newton’s mind seem and so worshipped had he been for two centuries.5

和汉密尔顿一样,德·摩根在还不到 22 岁时就担任了教授,当时他在英国第一所非宗派大学,即新成立的伦敦大学(不久后更名为伦敦大学学院)担任教授。两人出生时间相差不到一年,尽管德·摩根在婴儿时期就失去了一只眼睛。他曾因视力残疾而在学校受到欺负,但和欧拉一样,他没有让视力残疾阻碍自己成为一流的数学家。

Like Hamilton, De Morgan was not quite twenty-two when he’d first taken up a professorship—in his case, at England’s first nonsectarian university, the newly established London University, soon renamed University College, London. The two men were also born within a year of each other, although as a baby De Morgan had become blind in one eye. He’d been bullied at school for his ocular disability, but like Euler, he didn’t let it prevent him from becoming a first-rate mathematician.

德·摩根很幸运,他的妻子索菲亚·弗伦德是一位受过良好教育的社会活动家、辩论家和唯灵论者,与他志趣相投。他们都对自由充满热情,以至于他们采取了不同寻常的措施,在登记处结婚,以此来宣布他们的宗教独立。相比之下,汉密尔顿有着他的一位门徒所说的“深深的虔诚天性”,他是一个虔诚的英国国教徒,而他的妻子海伦据说特别虔诚。6

De Morgan was lucky enough to find a kindred soul in his wife, the well-educated social activist, polemicist, and spiritualist Sophia Frend. They were both so passionately committed to freedom that they’d taken the unusual step of getting married in a registry office, as a way of declaring their religious independence. By contrast, Hamilton had what one of his disciples would call “a deeply reverential nature,” and he was a committed Anglican, while his wife Helen was reputedly particularly religious.6

汉密尔顿也是一个情绪化的人,没有海伦在身边,她就无法轻松地工作。但有时——尤其是当她怀孕或哺乳时——她在距离都柏林五英里的邓辛克天文台黑暗偏僻的场地上感到焦虑,自从被任命为爱尔兰皇家天文学家以来,汉密尔顿就住在那里。她开始离家出走,住在附近的亲戚家,汉密尔顿经常和她住在一起。但是,在 1841 年——他们的女儿海伦·伊丽莎出生一年左右后——她离开了丈夫和孩子,和她在英国的姐姐住在一起。她离开了大约一年,各种各样的谣言开始四处流传,关于这场所谓的灾难性的婚姻,以及汉密尔顿似乎开始酗酒。他失去了他的初恋凯瑟琳·迪斯尼,这并没有帮助,因为当时严格的浪漫礼仪导致了误解,简·奥斯汀在十年或二十年前的小说中就强调了这一点。 19 世纪 30 年代和 40 年代,随着禁酒运动的兴起,人们对社交饮酒的态度急剧收紧,这也于事无补。因此,每过十年,这些故事就变得越来越夸张,不公平地玷污了汉密尔顿在世时的名声,直到今天仍然如此。7

Hamilton was also an emotional man and couldn’t work easily without Helen nearby. But sometimes—especially when she was pregnant or nursing—she felt anxious in the dark, remote grounds of Dunsink Observatory, five miles from Dublin, where Hamilton had lived since his appointment as Ireland’s royal astronomer. She began spending time away from home, staying with relatives nearby, and Hamilton often stayed with her there. But then, in 1841—a year or so after their daughter Helen Eliza was born—she left her husband and children to stay with her sister in England. She was away for about a year, and all sorts of rumours began to fly around, about the supposedly disastrous marriage and how Hamilton had apparently taken to drink. It didn’t help that he’d famously lost his first love, Catherine Disney, through the kind of misunderstandings engendered by the strict romantic etiquette of the day, which Jane Austen had highlighted in her novels just a decade or two earlier. It didn’t help, either, that attitudes to social drinking dramatically tightened in the 1830s and 1840s, as the temperance movement gathered pace. So, with each passing decade the stories became more and more exaggerated, unfairly tainting Hamilton’s reputation during his lifetime and continuing right up until today.7

虽然最近的研究表明,汉密尔顿夫妇的婚姻比传言中幸福得多,但德·摩根夫妇似乎过着幸福的家庭生活。索菲亚的父母是诗人拜伦夫人的朋友,拜伦夫人是这位诗人的离异妻子。1840 年至 1842 年的几年里,汉密尔顿夫妇正在应对海伦的焦虑和疾病,德·摩根则在辅导拜伦家的杰出女儿艾达·洛芙莱斯。(她的第一位数学导师是玛丽·萨默维尔,后来她搬到了气候更健康的意大利。)一年后,洛芙莱斯写了她对查尔斯·巴贝奇设计的先进计算机进行了数学开发——她的评论包括通常被认为是第一个计算机程序。它发表于汉密尔顿发现四元数的同一个月——事实证明,这是一个预言性的时间巧合,因为我们很快就会看到,正是计算机编程揭示了四元数的真正计算经济性。8

While recent scholarship suggests that the Hamiltons had a much happier marriage than the rumours suggest, the De Morgans seem to have had a charmed domestic life. Sophia’s parents were friends of Lady Byron, the poet’s estranged wife, and for a couple of years between 1840 and 1842, while the Hamiltons were dealing with Helen’s anxiety and ill health, De Morgan was tutoring the Byrons’ remarkable daughter, Ada Lovelace. (Her first mathematical mentor had been Mary Somerville, who had since moved to Italy for the healthier climate.) A year later, Lovelace wrote up her mathematical development of Charles Babbage’s design for an advanced calculating machine—and her commentary included what is often considered the first computer program. It was published in the very same month that Hamilton discovered quaternions—a prophetic coincidence of timing, as it turns out, for we’ll see shortly that it is computer programming that has brought to light the true computational economy of quaternions.8

与此同时,汉密尔顿仍在为寻找四元数而苦苦挣扎,当德摩根声称三维复数代数不可能时,汉密尔顿将其视为一个严肃的挑战。毕竟,他和德摩根都在寻找理解代数的方法——给它一个逻辑基础,就像两千年前欧几里得时代以来几何学所拥有的那样。正是这种欧几里得的严谨性让牛顿从几何上证明了《自然哲学的数学原理》的定理,而不是通过更紧凑但在方法论上仍然不稳定的代数微积分。至于更基本的代数问题,我已经指出,正是由于缺乏对负数和虚数的具体解释,早期的数学家们才放弃了它们——但正如德摩根在这种情况下讽刺地指出的那样,“没有什么比拒绝所有可能带来麻烦的东西更能让心灵放松的了。” 9

Meantime, Hamilton was still struggling in his quest for quaternions, and when De Morgan claimed the impossibility of a three-dimensional complex algebra, Hamilton had taken it as a serious challenge. After all, both he and De Morgan were searching for ways to make sense of algebra—to give it a logical foundation, the way geometry had had since the time of Euclid two thousand years earlier. It was this Euclidean rigour that had led Newton to justify the theorems of Principia geometrically, rather than via more compact but still methodologically shaky algebraic calculus. As for more fundamental algebraic problems, I’ve already indicated that it was the lack of a concrete explanation of negative and imaginary numbers that had led earlier mathematicians to dismiss them—but as De Morgan wryly noted in this context, “Nothing could make a more easy pillow for the mind, than the rejection of all which could give any trouble.”9

汉密尔顿绝对不是那种轻轻松松就能找到安乐窝的人。所以他一直埋头苦干,先尝试这个,然后尝试那个,从未放弃这个想法:如果你能用实数 ( x, y ) 来表示普通二维复数的代数,那么你就应该能够将其扩展为 ( x, y, z )——甚至,正如他在 1841 年告诉德·摩根的那样,可以扩展到任意维数,例如a = ( a 1 , a 2 , … , an ),其中a 1 , a 2 , … , an是实数。他很有先见之明地说,单个符号a “表示一个(复杂的)思想”。他继续从他的数学时间哲学的角度阐述这一点,向德·摩根建议,这n 个数字——你可以根据问题选择n的任意值——表示事件的时间顺序,因此是一种“因果关系的概念”。尽管汉密尔顿最终不再迷恋时间,但他还是瞥见了一些重要的东西:一个符号可以编码许多信息,或者“一个复杂的想法”。正如我所提到的,这是向量和张量的力量的一部分,它也有助于解释四元数的计算经济性。10

Hamilton was definitely not one to use an easy pillow. So he kept plodding away, trying first this and then that, and never giving up on the idea that if you could write the algebra of ordinary, two-dimensional complex numbers in terms of real numbers (x, y), then you should be able to extend it to (x, y, z)—and even, as he’d told De Morgan in 1841, to any number of dimensions, as in a = (a1, a2, … , an), where the a1, a2, … , an are real numbers. The single symbol a, he said presciently, is “indicative of one (complex) thought.” He went on to elaborate this in terms of his mathematical philosophy of time, suggesting to De Morgan that these n numbers—where you can choose any value of n, depending on the problem—represent a chronological ordering of events, and therefore a “notion of cause and effect.” But while Hamilton would ultimately move on from his fascination with time, he had glimpsed something important here: the idea that a single symbol can encode many pieces of information, or “one complex thought.” As I’ve mentioned, this is part of the power of vectors and tensors, and it will also help explain the computational economy of quaternions.10

汉密尔顿在探索过程中付出了巨大的努力,以至于他的书房里堆满了文件,散落在桌子上,堆在地板上,他经常不吃饭,靠带到房间里的零食维持生活。但后来有一天,他的坚持得到了回报,他突然有了一个惊人的想法。正如他向约翰·格雷夫斯解释的那样,“我突然想到,在某种意义上,我们必须承认空间存在第四维。”他的意思是,如果他引入第三个虚数k——这立即暗示了一个新的“4-D”复数a + ib + jc + kd的存在——并且如果他将k定义为ij,那么三元组的模数定律最终就会起作用。你可以在下面的尾注中看到原因,但正是这个奇妙的洞察力打动了他,就像“电路闭合火花闪现”,1843 年秋天那天,他和海伦在都柏林的布鲁姆桥上散步。11

Hamilton worked so hard in his quest that his study was a mess of papers, scattered over his desk and piled high on the floor, and he often skipped meals, surviving on snacks brought to his room. But then one day his persistence paid off, and suddenly he had a startling idea. As he explained to John Graves, “there dawned on me the notion that we must admit, in some sense, a fourth dimension of space.” What he meant was that if he introduced a third imaginary number, k—which immediately suggests the existence of a new “4-D” complex number a + ib + jc + kd—and if he defined k to be ij, then the law of moduli for triples would work at last. You can see why in the following endnote, but it was this marvelous insight that struck him, as if “an electric circuit closed and a spark flashed forth,” as he and Helen were walking by Broome Bridge on that autumn Dublin day in 1843.11

在命名他的新 4-D 数字时,我不知道他为什么不在“对”和“三元组”或“三重奏”后面加上“四元组”或“四重奏”——但“四元数”听起来确实很顺耳,而汉密尔顿对诗歌很有鉴赏力。正如德摩根后来回忆的那样,“汉密尔顿自己说过,‘我靠数学生活,但我是一个诗人。’这样的格言可能会让读者感到惊讶,但他们应该记住,数学发明的推动力不是推理,而是想象力。”然而,汉密尔顿的朋友华兹华斯认为,汉密尔顿的想象力更适合数学而不是诗歌。他确实非常有创造力,开创了矢量分析,而且根据量子先驱埃尔温·薛定谔的说法,他还开创了量子力学的核心数学。12

When it came to naming his new 4-D numbers, I’m not sure why he didn’t follow his “couples” and “triples” or “triplets” with “quadruples” or “quadruplets”—but “quaternion” does have a ring to it, and Hamilton had an ear for poetry. As De Morgan would later recall, “Hamilton himself said, ‘I live by mathematics, but I am a poet.’ Such an aphorism may surprise our readers, but they should remember that the moving power of mathematical invention is not reasoning, but imagination.” Hamilton’s friend Wordsworth, however, felt that Hamilton’s imagination was better directed toward mathematics than poetry. And he was indeed wonderfully inventive, pioneering both vector analysis and, according to no less an authority than quantum pioneer Erwin Schrödinger, the maths at the heart of quantum mechanics.12

汉密尔顿在看到这一惊人发现后非常兴奋,当场就在石桥上刻下了一条简单的线,

Hamilton had been so excited after his electric insight that right there and then he’d carved into the stone bridge a single, simple line,

i2 = j2 = k2 = ijk = −1

i2 = j2 = k2 = ijk = −1.

它包含了他所说的“用几何计算”所需的一切——包括计算三维空间中的旋转。为此简单方程式是一种简洁的方式来写下他的三个虚数之间的必要代数关系:

It contained all that he needed to “calculate with geometry,” as he put it—including calculating rotations in three-dimensional space. For this line of simple equations was a compact way of writing down the necessary algebraic relationships between his three imaginary numbers:

ij = k = − ji, jk = i = − kj, ki = j = − ik

ij = k = −ji, jk = i = −kj, ki = j = −ik.

(如果你用代数方法玩弄它,你就会从涂鸦中看到这一点:例如,k 2 = ijkk = ij。)13使用这些定义,如果你有耐心进行 4-D 乘法,例如

(You can see how this follows from the graffiti if you play with it algebraically: for instance, k2 = ijkk = ij.)13 Using these definitions, if you had the patience to carry out 4-D multiplications such as

( a + ib + jc + kd )( w + ix + jy + kz ),

(a + ib + jc + kd)(w + ix + jy + kz),

你会发现四元数不仅满足模数定律,还满足除交换律之外的所有普通实数和复数乘法定律。换句话说,

you’d see that quaternions satisfy not only the law of moduli, but all the laws of ordinary real and complex multiplication—except the commutative law. In other words,

( a + ib + jc + kd )( w + ix + jy + kz )≠( w + ix + jy + kz )( a + ib + jc + kd )。

(a + ib + jc + kd)(w + ix + jy + kz) ≠ (w + ix + jy + kz)(a + ib + jc + kd).

最后一个结论具有启示性,你无需进行繁琐的乘法运算就能猜到,因为虚数乘积本身(ij等)不具有交换性。对于汉密尔顿来说,这是一个不情愿的举动,他多年来一直试图将三元组和四元组纳入通常的算术法则,因此他值得称赞,因为他有足够的勇气迈出这一步,并认识到他已经发现了一种全新的代数。

It’s this last conclusion that was revelatory, and you can guess it without doing all the tedious multiplications, because the imaginary products themselves—ij and so on—are not commutative. This was a reluctant step for Hamilton, who had been working for years trying to fit first triples and then quaternions into the usual laws of arithmetic, so he deserves kudos for being brave enough to take it—and to recognise that he’d hit upon a brand-new kind of algebra.

非凡的全向量和四元数代数

THE REMARKABLE ALGEBRA OF WHOLE VECTORS AND QUATERNIONS

汉密尔顿将四元数分为两部分,例如P = w + ix + jy + kz。他将独立的实数称为“标量”,用w表示,将虚部ix + jy + kz称为“矢量”。矢量(vector)一词源于拉丁语,意为“载体”,半径矢量(radius vector)这一术语在天文学中用于表示将一个点“运送”到另一个点的可移动视线,比如从眼睛到恒星或行星。但正如汉密尔顿后来(在《四元数讲座》中)所指出的,“半径矢量”是一个标量,一个只表示量级的数,而他的术语“矢量”则表示量级方向。

Hamilton identified two parts to a quaternion such as P = w + ix + jy + kz. He called the real number standing by itself a “scalar,” represented here by w, and he named the imaginary part, ix + jy + kz, a “vector.” The word “vector” comes from the Latin for “carrier,” and the term “radius vector” had been used in astronomy for the movable line of sight that “carries” one point to another—say, from the eye to a star or planet. But as Hamilton would note later (in his Lectures on Quaternions), a “radius vector” is a scalar, a number representing only magnitude, whereas his term “vector” denotes magnitude and direction.

汉密尔顿的虚向量ix + jy + kz看起来很像你可能认识的向量x i + y j + z k,其中i、j、k以粗体显示,因为它们不再是基本虚数——而是实数“单位向量”,其大小或长度为“1”(即 1)。所以你已经可以看到当今向量的起源,尽管在我们了解这是如何发生的之前,故事还有很长的路要走。

Hamilton’s imaginary vector ix + jy + kz looks very like the vector you might recognise as xi + yj + zk, where the i, j, k are in boldface type because they are no longer fundamental imaginary numbers—rather, they’re real “unit vectors,” vectors whose magnitude, or length, is “unity” (which just means 1). So already you can see the genesis of today’s vectors, although there’s still a way to go in the story before we see how this happened.

四元数符号P编码了四部分信息——标量w和矢量的三个分量x、y、z——因此你可以将P视为代表“一个复杂的思想”,就像汉密尔顿在 1841 年写给德·摩根的焦急信中所预见的那样。

The quaternion symbol P encodes four pieces of information—the scalar w and the vector’s three components x, y, z—so you can think of P as representing “one complex thought,” just as Hamilton had foreseen in his anxious letter to De Morgan back in 1841.

如果我们从未来中抄袭粗体字,以区分汉密尔顿的矢量和他的标量,我们可以将四元数P写成,其虚数“基”为i, j, k,如下

If we crib the bold type from the future to distinguish Hamilton’s vector from his scalar, we can write the quaternion P, with its imaginarynumber “basis” i, j, k, as

P = w + p,其中p = ix + jy + kz

P = w + p, where p = ix + jy + kz.

(汉密尔顿有时把虚数放在实数分量之后,就像我们处理向量一样,因此ix, jy, kz中虚数和实数的顺序无关紧要 - 如果你愿意,可以将其视为xi, yj, zk。)“基”本质上只是一组独立的单位量,每个轴一个。更具体地说,这些基量被认为跨越进行计算的向量空间,但这些概念尚未得到正确的表述;这里的重点是,如果你从单位虚数基( i,j,k)更改为单位向量基(i,j,k ) ,也可以将信息写在向量p = ix + jy + kz中,并写为p = x i + y j + z k。这是因为关键在于在每种情况下,分量都有相同的数值(x,y,z),因为无论哪种方式你都使用笛卡尔坐标系。它是你用矢量进行计算时所需要的分量。

(Hamilton sometimes put the imaginary numbers after the real number components as we do with vectors, so the order of the imaginary and real numbers in ix, jy, kz doesn’t matter—you can think of it as xi, yj, zk if you prefer.) A “basis” is essentially just a set of independent unit quantities, one along each axis. More technically, these basis quantities are said to span the vector space in which the calculations take place, but these notions weren’t yet properly formulated; the point here is that you can write the information in the vector p = ix + jy + kz just as well if you change from the unit imaginary basis (i, j, k) to the unit vector basis (i, j, k), and write p = xi + yj + zk. That’s because the key thing is that in each case the components have the same numerical values, (x, y, z)—because either way you’re using a Cartesian coordinate system. And it is the components you need when you calculate with vectors.

换句话说,你可以将向量视为一种用于存储数据的设备(以组件的值形式存储),而基础则是设备的硬件。不同的设备具有不同的设计(在本例中是实数基础i、j、k或虚数基础i、j、k),但它们都存储相同的信息。不过,这只是事后的看法:从汉密尔顿向量到现代向量的令人惊讶的激烈转变是第 8 章的故事,但我在这里提到这一点是为了让熟悉现代向量的读者了解。

In other words, you can think of a vector as a device for storing data—in the values of the components—while the basis is the device’s hardware. Different devices have different designs—in this case a real basis, i, j, k or an imaginary one, i, j, k—but they both store the same information. This is hindsight, though: the surprisingly acrimonious transition from Hamilton’s vectors to modern ones is a story for chapter 8, but I mention it here for readers familiar with modern vectors.

如果你熟悉向量,那么汉密尔顿的四元数可能看起来很奇怪:又是一个苹果和橘子的例子,因为在本科向量分析中,我们不会把向量加到标量上。但在这个阶段,我们需要把向量简单地看作一个四维复数的虚部,就像汉密尔顿所做的那样——一个遵循他在布鲁姆桥上雕刻的虚数规则的代数表达式。

If you are familiar with vectors, Hamilton’s quaternion might seem strange: another case of apples and oranges, because adding a vector to a scalar is not something we do in undergrad vector analysis. But at this stage of the story, we need to think of a vector simply as the imaginary part of a 4-D complex number, as Hamilton did—an algebraic expression that follows the rules of the imaginary numbers he carved on Broome Bridge.

考虑到这一点,现在取第二个四元数,

With this in mind, now take a second quaternion,

Q = a + ib + jc + kd(或a + bi + cj + dk)= a + q

Q = a + ib + jc + kd (or a + bi + cj + dk) = a + q.

汉密尔顿发现,当你将PQ的乘积逐个分量相乘,并将所有得到的项分组为标量或矢量时,你会得到两种新的乘法,这里用现代的点和叉表示:

Hamilton discovered that when you multiply out the product of P and Q, component by component, and when you group all the resulting terms as scalars or vectors, you get two new kinds of multiplication, indicated here by the modern dot and cross:

PQ = wapq + wq + ap + p × q

PQ = wapq + wq + ap + p × q.

这是一个非常紧凑的表达式,因为它实际上包含二十二次乘法和十一次加法或减法。这种非凡的经济性是四元数和整个矢量(用特殊符号表示,比如这里的粗体,而不是列出它们的分量)威力的一个例子。除了减号之外,计算上汉密尔顿的“标量”和“矢量”积——分别是pqp × q ——与现代的完全相同;因此,为了了解四元数(和整个矢量)乘法的紧凑性,我在尾注中给出了这些乘积的现代分量形式,尽管你可能在数学课上熟悉它们。14你可能不知道,今天经常用于矢量的粗体字是由特立独行的英国人奥利弗·亥维赛 (Oliver Heaviside) 发明的。稍后我们会与彬彬有礼的美国人约西亚·威拉德·吉布斯 (Josiah Willard Gibbs) 一起正式会见他,他为我们提供了表示标量积的点 (现在也称为“点积”) 和表示矢量 (或叉) 积的叉。

This is an amazingly compact expression, given that it actually contains twenty-two multiplications and eleven additions or subtractions. This extraordinary economy is one example of the power of quaternions and whole vectors (the ones represented by special notation such as the boldface type here, rather than by listing their components). Aside from the minus sign, computationally Hamilton’s “scalar” and “vector” products— the pq and p × q, respectively—are just the same as the modern ones; so to see the compactness of quaternion (and whole-vector) multiplication, I’ve given the modern component forms of these products in the endnote, although you’re likely familiar with them from maths classes.14 But you may not know that the bold type often used today for vectors is due to the idiosyncratic Englishman Oliver Heaviside. We’ll meet him properly later, along with the urbane American Josiah Willard Gibbs, who gave us the dot for scalar products (also now called “dot products”) and the cross for the vector (or cross) product.

图像

图 4.1。交叉积的右手定则。如果你将右手的手指沿箭头所示方向从pq弯曲,在本例中为逆时针方向,那么你就可以从拇指的方向得到p × q的方向:它将指向上方或正z方向。要找到q × p的方向,你必须将右手倒置,顺时针弯曲手指,拇指将指向下方,即负z方向。在本例中,pq位于x - y(水平)平面,但无论它们位于哪个平面,它们的矢量积都将垂直于该平面。

FIGURE 4.1. Right-hand rule for cross products. If you curl the fingers of your right hand in the direction shown by the arrow from p to q, which in this case is anticlockwise, you get the direction of p × q from the direction of your thumb: it will point in the upward or positive z direction. To find the direction of q × p, you have to turn your right hand upside down to curl your fingers clockwise, and your thumb will point downward, in the negative z direction. In this case p and q are in the x-y (horizontal) plane, but no matter what plane they lie in, their vector product will be perpendicular to that plane.

汉密尔顿本人并没有将这两个新乘积指定为矢量的乘积;相反,他把它们称为元数乘积的“标量部分”和“矢量部分” 。然而,对于他的两个四元数PQ,汉密尔顿将标量wa设为零,并像我们今天一样将矢量分量相乘——只是他没有使用粗体字母和点或叉,而是将“标量部分”(我们的标量积)写为S.PQ ,将“矢量部分”(我们的矢量积)写为V.PQ。标量积是可交换的,但矢量积不是,这在称为“右手定则”的助记符中有所体现(图 4.1 )。这个规则的意思是,当你沿着从矢量p到矢量q 的方向弯曲手指时,你的拇指指向乘积矢量p × q的方向。但是如果你反转乘法的顺序,你的拇指就会指向相反的方向。这意味着p × q = − q × p。由于向量积是非交换的,所以整个四元数积PQ也是非交换的。

Hamilton himself hadn’t designated these two new products as products of vectors; rather, he called them the “scalar part” and the “vector part” of the quaternion product. Nevertheless, for his two quaternions P and Q, Hamilton set the scalars w and a to zero and multiplied out the vector components just as we do today—except that instead of using boldface letters and a dot or a cross he wrote S. PQ for the “scalar part” (our scalar product), and V. PQ for the “vector part” (our vector product). The scalar product is commutative, but the vector product is not, and this is encoded in the mnemonic device called the “right-hand rule” (fig. 4.1). What this rule means is that when you curl your fingers in the direction from vector p to vector q, your thumb points in the direction of the product vector p × q. But if you reverse the order of the multiplication, your thumb points in the opposite direction. Which means that p × q = −q × p. And because vector products are noncommutative, so is the whole quaternion product PQ.

正如您在图表(以及上一个尾注)中看到的,向量或叉积的关键在于它们将两个向量相乘得到另一个向量;标量或点积将两个向量相乘得到一个标量。(这就是为什么有一个数学笑话问,“当你把登山者和蚊子杂交时,你会得到什么?”答案是“什么也没有:你不能将标量与向量杂交”——因为叉积只能应用于两个向量,而不能应用于标量和向量。这也突出了“媒介”的生物学定义,即将病原体从一种动物或植物“携带”到另一种动物或植物的生物。)

As you can see in the diagram (and in the previous endnote), the key thing about vector or cross products is that they multiply two vectors to get another vector; scalar or dot products multiply two vectors to get a scalar. (Which is why there’s a maths joke that asks, “What do you get when you cross a mountain-climber with a mosquito?” The answer is, “Nothing: you can’t cross a scaler with a vector”—because the cross product can be applied only to two vectors, not to a scalar and a vector. This also highlights the biological definition of “vector,” meaning an organism that “carries” pathogens from one animal or plant to another.)

这种非交换向量乘法在抽象代数领域是一件大事,用汉密尔顿的话来说,这是一个“令人好奇、几乎疯狂”的发现。因为这是四千年有记载的数学史上第一次有人发现一个自洽的代数系统,其中XY 不等于YX。这意味着有一个全新的代数世界可供探索,谁知道有多少潜在的应用。这就是为什么汉密尔顿经常被描述为“代数的解放者”。

This noncommutative vector multiplication was a Very Big Deal in the world of abstract algebra, a “curious, almost wild” revelation, as Hamilton put it. For it was the first time in four thousand years of recorded mathematical history that someone discovered a self-consistent algebraic system where XY didn’t equal YX. And this meant there was a whole new algebraic world to explore, with who knew how many potential applications. That’s why Hamilton is often described as “the liberator of algebra.”

打破规则,开拓新世界

BREAKING THE RULES AND OPENING UP NEW WORLDS

汉密尔顿本人最初也对他的四元数打破了这一历史悠久的规则感到困惑和担忧。但在我们了解他如何用新数学来表示空间中的旋转之前,让我先向你展示一些其他非凡的新代数和应用,这些应用是由他发现非交换乘法而开辟的。

Hamilton himself was initially puzzled and concerned by the fact that his quaternions led him to break such a time-honoured rule. But before we see how he used his new mathematics to represent rotations in space, let me show you some of the other extraordinary new algebras and applications that his discovery of noncommutative multiplication opened up.

例如,如今非交换代数是量子力学和广义相对论的基础。在量子力学中,海森堡著名的不确定性原理指出,你不能同时精确测量量子粒子的位置X和动量P。但你可以先精确测量其中一个属性,比如位置,然后再测量另一个。问题是,当你进行第二次测量时,你第一次进行的位置测量将不再那么准确,因为亚原子粒子总是在移动或振动——所以如果你再次测量,你会得到不同的答案。这是非交换性在起作用,它有一个特殊的方程(海森堡原理由此得出,X 和 P 实际上是矩阵):

Today, for example, noncommutative algebras underpin quantum mechanics and general relativity. In quantum mechanics, Heisenberg’s famous uncertainty principle says that you cannot precisely measure a quantum particle’s position X and momentum P at the same time. But you can precisely measure first one of these attributes, say the position, and then the other. The trouble is that by the time you’ve made this second measurement, the position measurement you made first will no longer be quite so accurate, because subatomic particles are always moving or vibrating—so you’ll get a different answer if you measure it again. This is noncommutativity in action, and it has a special equation (from which Heisenberg’s principle follows, and X and P are, in fact, matrices):

=时长2π

XPPX=ih2π.

h是普朗克常数,它代表着测量基本极限,而且它非常小:以千克米2 /秒(或焦耳秒)为单位,为 6.6260693 × 10 −34 。无论它有多小, h都不为零,这就是关键所在——因为如果测量可观测位置和动量的顺序无关紧要,那么就会有XP − PX = 0。当然,这等同于XP = PX,而这正是交换律。

The h is Planck’s constant; it represents the fundamental limit to measurement, and it is tiny: 6.6260693 × 10−34 in units of kg m2/second (or joule-seconds). Tiny or not, h is not zero, and that’s the point here—for if it didn’t matter in which order you measured the observable position and momentum, you’d have XP − PX = 0. This, of course, is the same as XP = PX, which is none other than the commutative law.

在广义相对论中,非交换性给出了时空曲率的度量,因此当我们讨论张量时,我们会重新讨论这一点。同时,您可能已经想到,除了向量之外,普通矩阵也提供了一个更熟悉的非交换乘法示例。

In general relativity, noncommutativity gives a measure of the curvature of space-time, so we’ll revisit this when we come to tensors. Meantime, you might already be thinking that along with vectors, ordinary matrices offer a more familiar example of noncommutative multiplication.

《黑客帝国:重制版》(以及对当时技术的及时注释)

MATRIX REDUX (AND A TIMELY NOTE ON THE TECHNOLOGY OF THE TIME)

矩阵或数组是简洁地表示大量数据的绝佳方式,因此它们也与向量和张量有关。事实上,数学家们用数组和表格表示信息已有数千年的历史,但他们并没有将这些数组作为代数实体进行加法和乘法,直到亚瑟·凯莱(Arthur Cayley)阐明了如何做到这一点,即汉密尔顿(Hamilton)宣布四元数代数 15 年后。

Matrices, or arrays, are great ways of representing a lot of data concisely—so they, too, have something to do with vectors and tensors. Mathematicians had, in fact, been representing information in arrays and tables for thousands of years—but they hadn’t been adding and multiplying these arrays as algebraic entities in themselves until Arthur Cayley spelled out how to do it, fifteen years after Hamilton announced his quaternion algebra.

凯莱比汉密尔顿小 15 岁,生命中的前 8 年是在俄罗斯度过的(当时他的商人父亲住在圣彼得堡)。他以全班第一的成绩毕业于剑桥大学三一学院,并以临时奖学金在那里教了几年书。要成为正式的奖学金获得者,必须宣誓效忠圣公会,因此,尽管他是一个虔诚的圣公会教徒,但接受圣职还是太过分了,于是他在 1846 年开始接受律师培训。他几乎和欧拉一样多产,在担任房地产律师的 14 年里,他利用业余时间写了近 300 篇数学论文,之后又写了数百篇。他最早的一篇论文是在离开剑桥大学攻读法律之前写的,四元数——凯莱是第一个公开接受汉密尔顿思想的人。他于 1845 年发表了这篇论文,而汉密尔顿于 1843 年 11 月 13 日在爱尔兰皇家学院宣读的一篇论文中首次宣布了他的发现,不到一年。1848 年,汉密尔顿在都柏林大学就四元数做了四次系列讲座,凯莱当时也在场。15

Cayley, who was fifteen years younger than Hamilton, had spent the first eight years of his life in Russia (when his merchant father was based in St. Petersburg). He graduated top of his class at Trinity College, Cambridge, and taught there for several years on a temporary fellowship. A full fellowship required taking Anglican vows, so although he was a committed Anglican, taking Holy Orders was a step too far, and in 1846 he began training as a lawyer. He was almost as prolific as Euler, writing nearly three hundred mathematical papers in his spare time during the fourteen years he worked as a property lawyer as well as hundreds more afterward. One of his earliest papers, written before he left Cambridge for the law, was on quaternions—Cayley was the very first to publicly take up Hamilton’s idea. He published this paper in 1845, barely a year after Hamilton had first announced his discovery in a paper he read to the Royal Irish Academy on November 13, 1843. And when Hamilton gave a series of four lectures on quaternions at Dublin University in 1848, Cayley was in the audience.15

和所有事物一样,凯莱对矩阵的研究并非凭空而来,高斯和名气不大的费迪南德·爱森斯坦等数学家的研究都为其提供了先例。爱森斯坦在遇到汉密尔顿后,受到启发开始研究数学,但不幸在 29 岁时因肺结核英年早逝。凯莱自己对行列式的研究也有先例——行列式是在矩阵出现之前开发的,尽管今天我们将其作为矩阵的性质来了解。即使在古代,中国数学家也从他们提出和求解联立线性方程的方式中瞥见了矩阵的概念,他们使用数组和一种现在称为高斯消元法的形式。 (“线性”方程只有x、y等,就像直线方程一样——它们不包括未知数的乘积或幂,比如抛物线或圆的“二次”方程中的x2 除了描述直线的能力之外,线性方程还有许多有用的应用,比如优化——例如优化机器学习模型的准确性或优化业务成本。它们在描述如何以特定方式从一个坐标系转换到另一个坐标系时也至关重要——比如当你想物理旋转或平移笛卡尔系统或其中的向量时,正如我们稍后将在图 4.2中看到的那样。随着故事的进展,这种“线性变换方程”将发挥越来越重要的作用。)

Like everything, Cayley’s work on matrices didn’t arise in a vacuum, and there had been antecedents in the work of mathematicians such as Gauss and the lesser-known Ferdinand Eisenstein, who had been inspired to take up maths after he met Hamilton, but who died tragically early of tuberculosis when he was just twenty-nine. There were also antecedents in Cayley’s own work on determinants—which were developed before matrices, although today we learn about them as properties of matrices. Even in the ancient world, Chinese mathematicians had glimpsed the idea of matrices in the way they set out and solved simultaneous linear equations, using arrays and a form of what is now known as Gaussian elimination. (“Linear” equations have only x, y and so on, such as you get in the equations for straight lines—they don’t include products or powers of the unknowns, such as the x2 in the “quadratic” equation for a parabola, say, or a circle. Besides their ability to describe straight lines, linear equations have many useful applications, such as optimisation—optimising the accuracy of a machine learning model, for example, or optimising business costs. They are also crucial in describing how to change from one coordinate system to another in particular ways—such as when you want to physically rotate or translate your Cartesian system, or the vectors in it, as we’ll see later in figure 4.2. Such “linear transformation equations” will have an increasingly important role to play as this story progresses.)

古代中国的“高斯消元法”在两千年前的《九章算术》中得以保存,它远远领先于时代。中世纪印度数学家也做了类似的事情——公元七世纪,婆罗门笈多甚至使用数组来求解二次方程。所有这些早期方法都是纯算法的——非常适合计算机!因此,它们是专门计算的——对每个问题都有一个隐含的“先这样做,然后那样做”的计划——而不是像符号矩阵代数那样的代数和通用方法(或者实际上像牛顿和莱布尼茨的符号微积分算法)。因为高斯消元法不使用矩阵乘法或加法。16凯莱使用的和《九章论》那位不知名的作者偶然发现的联立方程的方便表示法相同——但是因为他受益于符号代数,这最终使他走向了矩阵理论。不过,一开始,他只是想像他的前辈一样解方程——只不过他研究的是他的朋友詹姆斯·西尔维斯特很快就称之为“不变量”的东西。17稍后我们会更多地讨论不变量,因为它们是张量概念的基础,在相对论中尤其重要。它们还使用我刚才提到的线性变换方程的概念。

The ancient Chinese “Gaussian elimination” method survives in the two-thousand-year-old Jiuzhang Suanshu, or Nine Chapters on the Mathematical Art, and it was way ahead of its time. Medieval Indian mathematicians did something similar—and in the seventh century Brahmagupta even used arrays to solve quadratic equations. All these early approaches were purely algorithmic—ideal for a computer! So, they were specifically computational—with an implied “do this then that” plan for each problem—rather than algebraic and general like symbolic matrix algebra (or, indeed, like Newton’s and Leibniz’s symbolic calculus algorithms). For Gaussian elimination doesn’t use matrix multiplication or addition.16 Cayley was using the same kind of convenient representation of simultaneous equations that the unknown author of Nine Chapters had hit upon— but because he had the benefit of symbolic algebra, this eventually led him to the theory of matrices. To begin with, though, he was simply trying to solve equations, just as his ancient predecessors had done—except that he was studying what his friend James Sylvester would soon dub “invariants.”17 More on invariants later, for they are fundamental to the idea of a tensor, and they’re especially important in relativity theory. They also use the idea of linear transformation equations that I just mentioned.

19 世纪 40 年代初,凯莱开始与乔治·布尔通信,讨论变换和不变量的问题。今天,我们因布尔发展了计算机的基础“布尔”代数逻辑而想到布尔,但布尔的出现稍晚一些,当时汉密尔顿的四元数已经开启了全新的符号代数的可能性。这是四元数的发现不仅对它本身很重要,而且有助于激发一系列新的代数发现,从矩阵到符号逻辑。无论如何,凯莱当时还是剑桥大学一位年轻的数学教授,而自学成才的布尔在林肯管理着一所学校,尽管他很快就获得了科克的教授职位。两人就他们的研究交换了信件,凯莱开始希望他们能亲自见面——但他遗憾地发现,这两个城镇之间还没有铁路连接。18

In the early 1840s, Cayley began corresponding on the subject of transformations and invariants with George Boole. Today we think of Boole for his development of the “Boolean” algebraic logic that underpins computers, but that came a little later, after Hamilton’s quaternions had opened up the possibility of completely new symbolic algebras. It’s another example of the way the discovery of quaternions was important not only for itself, but also for helping to inspire a whole range of new algebraic discoveries, from matrices to symbolic logic. Anyway, Cayley was still a young professor of mathematics at Cambridge, and the self-taught Boole was running a school in Lincoln, although he would soon gain a professorship in Cork. The two men exchanged letters on their research, and Cayley began to wish they could meet in person—but, he lamented, there was not yet a rail link between the two towns.18

凯莱的哀叹是对当时技术的一个有趣评论——提醒人们,第一条使用蒸汽机车运载乘客和货物的公共铁路才建成不到十年,而且还在扩建。然而,它们已经为纺织品等制成品创造了新的市场,而这反过来又推动了新的科学和技术创新。工业革命正在顺利进行——蒸汽机的发明就是其中的一部分——铁路项目本身也引发了新科学发现的新应用,例如使用全新的电磁电报发明来协调列车交通。电磁学的存在仅在二十年前才被发现,因此该应用强调了一方面经济学和企业家精神与另一方面科学、数学和技术之间的古老双向联系。

Cayley’s lament is an interesting comment on the technology of the times—a reminder that the first public railways using steam locomotives to carry passengers as well as goods had been built barely a decade earlier, and they were still being expanded. Yet already they were creating new markets for manufactured products such as textiles, which in turn would drive new scientific and technological innovations. The Industrial Revolution was well underway—the invention of the steam engine had been part of it—and the railway project itself was sparking new applications of new scientific discoveries, such as the use of the brand-new invention of electromagnetic telegraphy to coordinate train traffic. The very existence of electromagnetism had only been discovered twenty years earlier, so this application highlights the ancient, two-way connection between economics and entrepreneurship on the one hand and science, mathematics, and technology on the other.

不幸的是,这种联系常常忽视技术对环境的影响。当时,至少还有一个借口,如果无知也算借口的话:又过了 10 年,美国气候学家先驱尤妮丝·牛顿·富特(Eunice Newton Foote,艾萨克爵士的远亲)发表了关于大气二氧化碳加热效应的里程碑式论文——这项令人兴奋的 19 世纪新技术大肆宣扬这一点。富特和其他早期气候科学家并没有过多考虑未来的气候变化;他们关注的是地球过去的气候是什么样的,以及如何推断出现在的温度。19然而,有些人对这种“进步”感到不安——最著名的例子可能是美国作家亨利·戴维·梭罗,1845 年,他隐居在马萨诸塞州康科德市瓦尔登湖畔的森林中,以便更贴近大自然。与他的诗人朋友拉尔夫·沃尔多·爱默生一样,他受到超验主义、新英格兰哲学和文学回归自然运动的启发——与汉密尔顿交往的英国浪漫主义诗人有某种大致的相似之处。

Unfortunately, it’s a nexus that too often ignores the environmental impact of technology. Back then, at least there was an excuse, if ignorance counts as an excuse: it would be another decade before the pioneering American climate scientist Eunice Newton Foote, a distant relative of Sir Isaac, published her landmark paper on the heating effect of atmospheric carbon dioxide—which much of this exciting new nineteenth-century technology was spouting profusely. Not that Foote and other early climate scientists were thinking much about future climate change; they were focussed on what Earth’s climate was like in the past, and how to deduce its present temperature.19 Some, however, were uneasy about this kind of “progress”—perhaps most famously the American writer Henry David Thoreau, who in 1845 withdrew to the woods by Walden Pond in Concord, Massachusetts, in order to live closer to nature. Like his poet friend Ralph Waldo Emerson, he was inspired by transcendentalism, New England’s philosophical and literary back-to-nature movement—a kind of loose parallel to the English Romantic poets with whom Hamilton associated.

不过,凯莱并没有考虑工业与科学的联系——他从事数学是因为热爱它。因此,他很快意识到矩阵不仅仅是表示方程的便捷方法——它们本身就是代数结构,有自己的代数规则,这并不奇怪。

Cayley wasn’t thinking about the industrial nexus with science, though—he was engaged in mathematics for the love of it. So it’s not surprising that he soon realised matrices were more than just convenient methods of representing equations—they were algebraic structures in their own right, with their own algebraic rules.

例如,使用代数符号,你可以写出联立线性方程

For example, using algebraic symbolism you can write the simultaneous linear equations

2x + y =

7x3y = 1

2x + y = 7

x − 3y = 1

矩阵形式如下:

in matrix form like this:

2113=71

2113xy=71

(顺便一提,71是“列向量”的例子——其中信息写成列而不是我到目前为止一直使用的行,例如(x,y)。无论哪种方式,信息都是相同的。)这样写这两个方程式需要“逐列”矩阵乘法的想法,但凯莱最初是在尝试寻找一种简洁、省力的方式来表示多个线性变换时,想到了将任何两个兼容矩阵相乘——而不仅仅是一个矩阵和一个向量。例如,如果要旋转坐标系然后平移轴,则需要变换方程来从x - y坐标旋转到x′ - y′坐标(正如我们很快就会在图 4.2中看到的那样)。然后,您需要从x′ - y′变换到x″ - y″才能平移轴。事实证明,将这些变换组合起来得到x - y (以x″ - y″表示)时,最终矩阵中的元素给出了逐行定义。而这个定义又是乘法不总是可交换的另一种情况。

(By the way, xy and 71 are examples of “column vectors”—where the information is written as a column rather than the rows I’ve been using so far, such as (x, y). It’s the same information either way.) Writing these two equations like this takes the idea of “row-by-column” matrix multiplication for granted, but Cayley initially got the idea for multiplying any two compatible matrices—not just a matrix and a vector—when he was trying to find a neat, labour saving way of representing multiple linear transformations. For example, if you want to rotate a coordinate frame and then translate the axes, you’ll need transformation equations to rotate from x-y coordinates to x′-y′ coordinates, say (as we’ll see shortly, in fig. 4.2). Then you’ll need to transform from x′-y′ to x″-y″ in order to translate the axes. It turns out that on combining these transformations to get x-y in terms of x″-y″, the entries in the final matrix give the row-by-column definition. And this definition turns out to be another case where multiplication is not always commutative.

换句话说,一组传统线性方程中的信息可以以矩阵形式表示,就像古代中国数学家所看到的那样,但在凯莱手中,这导致了这样的想法:矩阵本身可以相乘,而不管它们代表什么。因此,他认为它们也可以相加:要添加两个矩阵(大小相同),只需在相应的位置添加两个数字,如下所示:

In other words, the information in a set of conventional linear equations can be represented in matrix form, as the ancient Chinese mathematicians had seen, but in Cayley’s hands this led to the idea that matrices themselves could be multiplied, regardless of what they represented. So, he figured that they could be added, too: to add two matrices (of the same size), you just add the two numbers at the corresponding places, like this:

3120+1526=4406

3120+1526=4406

凯莱的密友兼同事西尔维斯特也是一位数学天才律师,他在 1850 年创造了“矩阵”一词——恰巧,同年,第一条国际电报电缆在多佛和加莱之间铺设。你可以想象到人们兴奋地谈论通过电报将世界统一起来——就像我们最近在互联网的连接能力上看到的创业炒作和真正的兴奋一样。(不幸的是,多佛电缆失败了——但很快其他电缆也失败了。)西尔维斯特和凯莱一样,曾在剑桥学习,但作为犹太人,他没有资格毕业,因为教派规定导致德摩根拒绝在那里任教。因此,西尔维斯特和凯莱以律师的身份相识——但他们都致力于不变理论和他们利用业余时间研究矩阵,最后都成为了数学教授,追寻着自己真正的使命。

Cayley’s close friend and colleague Sylvester, yet another mathematically gifted lawyer, had coined the term “matrix” in 1850—the same year, as it happens, that the first international telegraphic cable was laid, between Dover and Calais. You can imagine the excited talk about uniting the world via telegraphy—the same mix of entrepreneurial hype and genuine excitement we’ve witnessed more recently, over the connective power of the internet. (Unfortunately, the Dover cable failed—but others soon followed.) Sylvester, like Cayley, had studied at Cambridge, but as a Jew he wasn’t entitled to graduate, thanks to the sectarian regulations that had led De Morgan to refuse to teach there. So, Sylvester and Cayley met in their capacity as lawyers—but they both worked on invariant theory and matrices in their spare time, and they both ended up as maths professors, following their true calling.

虽然凯莱被普遍认为是矩阵代数的创始人,但其他数学家很快就开发出了更为复杂的矩阵理论。例如,在美国,父子数学家本杰明和查尔斯·皮尔斯是矩阵和许多其他新后四元数代数的先驱,因为汉密尔顿打开了代数闸门。不过,凯莱寻找矩阵代数规则的动机只是为了更有效地进行计算,这一点很能说明问题:就像在古代一样,许多新的数学被创造出来是为了解决实际问题或促进计算。但是,一旦你可以在一个环境中乘以矩阵,它们就可以像向量一样应用于大量其他实际计算。

While Cayley is generally credited as the founder of matrix algebra, other mathematicians soon developed more sophisticated matrix theory. In America, for example, father and son mathematicians Benjamin and Charles Peirce were pioneers of matrix and many other new, post-quaternion algebras, for Hamilton had opened the algebraic floodgates. Still, it is telling that Cayley had been motivated to find the rules of matrix algebra simply to carry out computations more efficiently: just as in ancient times, a lot of new maths is created to solve practical problems or to facilitate computation. But once you can multiply matrices in one context, they, like vectors, can be applied to an amazing range of other practical computations.

图像压缩、搜索引擎、机器学习、机器人:矩阵和向量乘法的应用

IMAGE COMPRESSION, SEARCH ENGINES, MACHINE LEARNING, ROBOTICS: APPLICATIONS OF MATRIX AND VECTOR MULTIPLICATION

一旦知道如何乘以矩阵,有时就可以“分解”(或“分解”)它们。其中一种分解方法称为“奇异值分解”(SVD),其现代用途之一是数字图像压缩。图像中要传输的信息(例如每个像素的位置和颜色)表示为具有数百或数千行和数千列的矩阵。然后将图像矩阵分解为三个矩阵,其中一个因子(“奇异”因子)包含图像中不太有趣的部分(例如天空或其他背景)。简单地说,这些不太有趣的细节可以“分解”出来(也是借助向量),因此要传输的信息要少得多。

Once you know how to multiply matrices, sometimes you can “factorise” (or “decompose”) them. One such factorisation is called the “singular value decomposition” (SVD), and one of its modern uses is in digital image compression. The information—such as the location and colour of each pixel—in the image you want to transmit is represented as a matrix with hundreds or thousands of rows and columns. The image matrix is then factorised into three matrices, with one of these factors (the “singular” one) containing the less interesting parts of the image—such as the sky or other background. Speaking very loosely, this less interesting detail can then be “factored out”— with the help of vectors, too—so there’s much less information to transmit.

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• • •

说到海量信息,在搜索引擎编程中,可以通过矩阵和向量代数紧凑地处理大量数据。最早的搜索引擎方法之一是使用布尔代数通过布尔的 AND、OR 和 NOT 运算为用户的查询提供精确匹配。但正如康奈尔大学计算机科学教授 Gerry Salton 意识到的那样在 20 世纪 60 年代,矩阵和向量允许部分匹配用户的查询并根据相关性进行排序。(后来,布尔搜索引擎也引入了模糊布尔逻辑来实现这一点,每种方法都有其优点和缺点。)在向量方法中,信息以矩阵的形式存储,行表示关键词,列表示包含这些关键词的不同文档。为了简化问题,我假设数据库只有三个关键词,我将其称为 A、B 和 C。文档 1 可能不包含对关键词 A 和 B 的引用,但包含对 C 的三个引用,因此它将表示为向量003,矩阵的第一列。文档 2 可能对 A 有一次引用,对 B 有三次引用,对 C 有两次引用,因此它将表示为132等等。现在假设用户想要搜索与 B 相关的信息,那么“查询向量”可以写成010. 你用 0 表示它们,而你想要的是 B,用 1 表示。)由于每个文档都用一个向量表示,因此它与查询向量的相关性可以通过标量积(几何定义,如下一个尾注)从其向量相对于查询向量的角度来计算;角度越小,匹配越接近。20

Speaking of huge amounts of information, in programming search engines, large arrays of data can be handled compactly via matrix and vector algebra. One of the earliest search engine methods used Boolean algebra to give exact matches to a user’s query, via Boole’s AND, OR, and NOT operations. But as Cornell computer science professor Gerry Salton realised in the 1960s, matrices and vectors allow users’ queries to be matched partially and ranked according to relevance. (Fuzzy Boolean logic was later introduced in Boolean search engines to do this, too, and each approach has its advantages and disadvantages.) In the vector approach, information is stored as a matrix, with rows representing, say, key words, and columns representing different documents containing these key words. To simplify matters I’ll assume the database has just three key words, which I’ll call A, B, and C. Document 1 might contain no references to keywords A and B, say, but three references to C, so it would be represented as the vector 003, the first column of the matrix. Document 2 might have one reference to A, three to B, and two to C, so it would be represented as 132, and so on. Now suppose the user wants to search for information relating to B, so the “query vector” can be written as 010. you represent them by 0, and you do want B, denoted by 1.) Since each document is represented by a vector, its relevance to the query vector can be computed, via the scalar product (defined geometrically, as in the next endnote), from its vector’s angle with respect to the query vector; the smaller the angle, the closer the match.20

另外,在谷歌的 PageRank 算法中(该算法由斯坦福大学研究生拉里·佩奇和谢尔盖·布林于 20 世纪 90 年代末开发),网站根据链接到它的其他网站的数量和排名进行分类。表示从网站 A 到其他每个网站的链接比例的向量是矩阵的第一列(我将其称为M),其他网站也是如此。要找到每个网站本身的排名,最好先对所有网站进行相同的排名。然后通过将这个初始排名向量乘以链接偏好矩阵来更新它。然后这个过程不断重复,每次都乘以M,这样到第n步时,初始排名向量已经乘以了M n。这个过程不断迭代,直到排名向量不再有显著变化。在这个“平衡”点,网站的 PageRank 向量被定义。随着更多网站链接的加入,这个过程不断更新,正如你在搜索某些内容时看到的那样,并注意到哪些网站首先出现。但如果没有矩阵乘法的定义,这一切都不可能实现。21

Alternatively, in Google’s PageRank algorithm—developed in the late 1990s by Stanford grad students Larry Page and Sergey Brin—a website is classified according to the number and rank of the other websites that link to it. The vector representing the proportionate number of links from website A to each of the other websites is the first column of a matrix (which I’ll call M), and so on for the other websites. To find the rank of each website itself, it makes sense to start off with an equal ranking for all of them. This initial rank vector is then updated by multiplying it by the matrix of linking preferences. The process then keeps repeating, each time multiplying by M, so that by the nth step, the initial rank vector has been multiplied by Mn. The process keeps on iterating until there’s no significant change in the rank vector. At this “equilibrium” point the website’s PageRank vector is defined. With more website links coming on board, the process keeps on updating, as you can see when you search for something and note which websites come up first. But none of this would be possible without a definition of matrix multiplication.21

• • •

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在机器学习中,信息也存储在字符串和数组(向量和矩阵,甚至多维数组(实际上是张量的例子))中。在现实世界中,一般只能收集有关某个主题的所有可能数据的样本。数据科学包括从该样本推断以进行预测(比如明年某人心脏病发作的可能性,或在特定时间的可能负荷和能源价格)和分类(比如肿瘤是恶性的还是良性的,电子邮件是有效的还是垃圾邮件,或者最近引起争议的一张脸是否是你的脸)。程序员通过将数据拟合到“模型”中来实现这一点——一种可以理解数据的算法或方程。但有时设计一个适当准确的模型来拟合手头的数据太困难或成本太高,这就是机器学习的作用所在。这个想法是用一种算法对计算机进行编程,通过一个简单的初始模型“训练”计算机进行预测或分类;输入初步“训练”数据、初始模型和错误测试程序。利用向量和矩阵代数,程序员告诉计算机如何估计初始模型的准确性,并不断调整它,直到达到最佳准确性——类似于 PageRank 算法中的矩阵乘法迭代。然后模型就被“训练”了,程序员现在可以使用它根据新数据进行预测。

In machine learning, too, information is stored in strings and arrays— vectors and matrices, and even multidimensional arrays (which are, in fact, examples of tensors). In the real world it is generally only possible to collect a sample of all the possible data about a topic. Data science includes the art of extrapolating from this sample to make predictions—such as the likelihood of a person having a heart attack in the next year, or the likely load and price of energy at a given time—and classifications, such as whether a tumor is malignant or benign, whether an email is valid or spam, or, more recently and controversially, whether a face is yours or not. Programmers do this by fitting the data to a “model”—an algorithm or an equation that makes sense of the data. But sometimes it is too difficult or costly to devise an appropriately accurate model to fit the data at hand, and this is where machine learning comes into it. The idea is to program the computer with an algorithm that “trains” it to make predictions or classifications via a simple initial model; preliminary “training” data are fed in, along with the initial model and an error-testing routine. Making use of vector and matrix algebra, the programmer tells the computer how to estimate the accuracy of the initial model and to keep readjusting it until it reaches an optimum accuracy—analogously to the matrix multiplication iterations in the PageRank algorithm. The model has then been “trained,” and the programmer can now use it to make predictions from new data.

机器学习发展如此迅速,以至于它已经渗透到了日常生活中令人眼花缭乱的各个方面,从聊天机器人和推荐算法到语音和图像识别、欺诈检测、自动驾驶汽车、医疗诊断等等,有时是好的,有时是坏的。(“坏的”不仅包括专制政府或网络犯罪分子对人工智能的滥用,以及对我们生活和工作方式的颠覆性改变,还包括围绕用于训练人工智能的数据的道德问题。这些问题包括未经创建者许可使用数据,以及程序员(主要是白人男性)无意识地将种族、性别和性别偏见编入训练数据、搜索引擎和其他算法中。22 )

Machine learning has expanded so rapidly that it is now behind a dizzying range of everyday things, from chatbots and recommendation algorithms to speech and image recognition, fraud detection, self-driving cars, medical diagnostics, and much more—for better and sometimes worse. (The “worse” includes not only misuse of AI by repressive governments or cyber criminals, and the disruptive changes to the way we live and work, but also the ethical issues surrounding the data used to train AI. These issues include using data without permission from their creators, and the racial, sexual, and gender biases that programmers—largely white men—have unconsciously programmed into training data, search engines, and other algorithms.22)

矩阵有无数其他用途——包括密码学,其中矩阵乘法(以及其他代数结构)的非交换性目前正在被探索作为一种额外的安全工具。这项研究的先驱之一是一位 16 岁的爱尔兰女学生莎拉·弗兰纳里。她非常钦佩凯利,以至于她用他的名字命名了她的算法。23

Matrices are used in countless other ways—including cryptography, where the noncommutativity of matrix multiplication (and of other algebraic structures) is currently being explored as an additional safety tool. One of the pioneers in this research was a sixteen-year-old Irish schoolgirl, Sarah Flannery. She admired Cayley so much that she named her algorithm after him.23

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不过,现在,如果你已经学习过线性代数课程,你可能会认为矩阵乘法的一个更简单、更古老、更熟悉的应用是旋转——你是对的!所以,我们开始关注汉密尔顿对旋转的关注。

By now, though, if you’ve taken a linear algebra course, you might be thinking that a simpler, older, and more familiar application of matrix multiplication is rotations—and you are right! So, we are beginning to home in on Hamilton’s preoccupation with rotations.

例如,在机器人技术中,程序员需要知道如何编写指令,以便计算机可以告诉机器人如何从一个地方移动到另一个地方。例如,要抬起和放下机器人手臂,必须绕铰链旋转它(就像肩关节或肘关节一样),如图4.2所示。在计算机图形学中,关键特征的位置向量也可以通过旋转矩阵在空间中移动。

In robotics, for example, programmers need to know how to write instructions so that the computer can tell the robot how to move from one place to another. To raise and lower a robot arm, for instance, it must be rotated about a hinge (just like a shoulder or elbow joint), as in figure 4.2. In computer graphics, too, the position vectors of key features can be moved around in space via rotation matrices.

图像

图 4.2。使用旋转矩阵将机械臂绕xy平面的原点逆时针旋转角度 θ。当您乘以这个矩阵方程的左边时,得到的方程显示了如何用x和y表示x y′,表示线性变换。不过,细节在这里并不那么重要:关键是变换方程的概念

FIGURE 4.2. Rotating a robot arm about the origin of the xy plane, anticlockwise through an angle θ, using a rotation matrix. When you multiply out the left-hand side of this matrix equation, the resulting equations, showing how to write x′ and y′ in terms of x and y, represent a linear transformation. The details are not so important here, though: key is the idea of transformation equations.

最终实现 3-D 旋转!

3-D ROTATIONS AT LAST!

对于三维空间中的旋转,矩阵乘法当然比图 4.2中稍微复杂一些。但与四元数一样,矩阵的非交换乘法也非常适合这项工作,因为三维空间中的两个连续旋转通常也不交换。用一本书试试看:把它放在桌子上,封面朝上,书脊朝向你。然后,保持书的中心固定,将书顺时针旋转 90°,使书脊在你的左边(这是在桌子水平面上的旋转,比如说x - y平面)。现在将书顺时针翻转,旋转 180°,直到它正面朝下放在桌子上。这一次,你在垂直的x - z平面上旋转,因此你已经使用了空间的所有三个维度——当你完成后,你会注意到书的书脊在你的右边。现在重新开始,但先把它翻过来,然后朝你自己的方向旋转:脊柱就在你的左边!

For rotations in three-dimensional space, the matrix multiplications are a little more complicated than in figure 4.2, of course. But as with quaternions, too, matrices’ noncommutative multiplications are right for the job, because two successive rotations in 3-D space generally don’t commute, either. Try it with a book: Place it on a table with the front cover facing up and the spine toward you. Then, keeping its centre fixed, rotate the book clockwise through 90°, so that the spine is on your left (this is a rotation in the horizontal plane of the table, say the x-y plane). Now turn the book over clockwise, rotating it through 180° until it’s face down on the table. This time, you’re rotating in the vertical x-z plane, so you’ve used all three dimensions of space—and when you’re done, you’ll notice that the book’s spine is on your right. Now start again, but turn it over first, and then rotate it toward you: the spine will be on your left!

顺便说一句,如果你用一个没有特征的方形盒子或一个没有标记的球来做实验,你将无法辨别这种非交换性。这是因为在这样的旋转中,球体和立方体会保持它们的形状和方向,因为它们是围绕中心对称的。换句话说,它们在这种旋转序列下是“不变的”。这是凯莱在提出矩阵代数时研究的那种不变量的几何类比。

By the way, if you were to do the experiment with a featureless square box, or a ball with no markings, you wouldn’t be able to discern this noncommutativity. That’s because with rotations like these, spheres and cubes keep their shape and orientation on account of their symmetry about their centres. In other words, they are “invariant” under this sequence of rotations. This is a geometrical analogue of the kind of invariants that Cayley was studying when he came up with matrix algebra.

比凯莱早一个世纪左右,欧拉利用古希腊人开创的球面三角学,对二维和三维旋转数学进行了出色的几何研究。另一方面,矩阵代数则得益于四元数及其非交换乘法——四元数最终使汉密尔顿能够给出空间旋转的第一个纯代数计算。

Almost a century before Cayley, Euler had made a brilliant geometrical study of two- and three-dimensional rotation mathematics, using the spherical trigonometry pioneered by the ancient Greeks. Matrix algebra, on the other hand, is indebted to quaternions and their noncommutative multiplications—and quaternions had enabled Hamilton to give, at last, the very first purely algebraic computations for rotations in space.

首先,为了表示想要旋转的三维矢量,他只取四元数的矢量部分 - 例如,在P = w + ix + jy + kz中设置w = 0 ,这样他就只得到了矢量p = ix + jy + kz

First up, to represent the three-dimensional vector he wanted to rotate, he took just the vector part of a quaternion—for example, by setting w = 0 in P = w + ix + jy + kz, so that he had just the vector p = ix + jy + kz.

图像

图 4.3。旋转复数的“机制”(参见图 3.8 )。用复数表示点A

FIGURE 4.3. The “machinery” for rotating a complex number (cf. fig. 3.8). Represent the point A by the complex number

x + iy = r cos θ + ir sin θ = rei θ

x + iy = r cos θ + ir sin θ = reiθ.

要将OA绕原点旋转 α 角,将其乘以单位复数b = ei α得到新点

To rotate OA about the origin through an angle of α, multiply it by the unit complex number b = eiα to give the new point

B = ( ei α )( x + iy ) = ( ei α ) re i θ = rei (θ+α)

B = (eiα)(x + iy) = (eiα)reiθ = rei(θ+α).

(如果只想旋转原始向量,而不改变其长度,则“机械”必须是单位向量,如图3.8所示。)

(The “machinery” has to be a unit vector if you just want to rotate the original vector, without changing its length as well, as in fig. 3.8.)

然后,对于进行旋转的代数“机械”,他选择了一个单位四元数U,对所需的旋转轴和角度进行编码。这类似于图 4.3中的单位复数

Then, for the algebraic “machinery” that will do the rotating, he chose a unit quaternion U encoding the required axis and angle of rotation. It’s analogous to the way that in figure 4.3, the unit complex number

b = 余弦 α + i正弦 α

b = cos α + i sin α

是将直线OA绕原点在二维复平面上旋转的“机制”。类似地,要在三维空间中进行旋转,必须将三维向量p乘以四维四元数U——因此,汉密尔顿耐心寻找将i、jk相乘的正确规则终于有了回报。其余机制都已到位——我已经下一个尾注中描述了这一点——一些巧妙但简单的计算给出了旋转后的向量。尾注显示了当pi轴旋转时如何做到这一点,你可以在图 4.4看到结果。24

is the “machinery” that rotates the line OA about the origin in the 2-D complex plane. Similarly, to get a rotation in three-dimensional space, the 3-D vector p has to be multiplied by the 4-D quaternion U—so this is where Hamilton’s patient search for the right rules for multiplying his i, j, and k finally paid off. With the rest of the machinery in place—which I’ve described in the next endnote—some nifty but simple calculations give the rotated vector. The endnote shows how to do this when rotating p about the i-axis, and you can see the result in figure 4.4.24

图像

图 4.4 . 使用四元数的三维旋转。当p = ix + jy + kzi轴旋转 2θ 角时,其新位置向量为a = xi + ei ( yj + zk )。我选择 2θ 作为角度而不是 θ 是因为它与量子力学有着有趣的联系,我们很快就会看到。

FIGURE 4.4. 3-D rotations using quaternions. When p = ix + jy + kz is rotated through an angle of 2θ about the i-axis, its new position vector is a = xi + ei(yj + zk). I’ve chosen 2θ for the angle rather than θ because it will have an interesting connection with quantum mechanics, as we’ll see shortly.

在数学课上,您可能会发现使用矩阵的旋转向量,正如我所提到的,这是一种很好的方法。但事实证明,在更复杂的情况下,四元数比矩阵更有效,因为您需要组合围绕不同轴的几次旋转,以便以特定方式定位物体或通过平滑的旋​​转序列来驱动它 - 例如卫星、飞机、机器人、手机屏幕或计算机动画。四元数比矩阵更紧凑,而且它们更快,因为它们需要的计算更少,并且消耗的计算机功率更少。

In maths classes, you would likely have found a rotated vector using matrices, which is a good way to do it, as I’ve mentioned. But it turns out that quaternions are more efficient than matrices in more complicated situations where you need to combine several rotations around different axes, in order to orient an object in a particular way or to drive it via a smooth sequence of rotations—a satellite, for instance, or an airplane, a robot, your mobile phone screen, or a computer animation. Quaternions are more compact than matrices, and they are faster, because they require fewer calculations and use less computer power.

例如,在对飞机或航天器进行建模、模拟、跟踪或引导时,其方向(或“姿态”)是从三个垂直轴测量的——通常,沿着飞行器的长度、宽度或翼展以及垂直上下方向,原点位于重心。围绕这些轴的旋转分别称为“滚动”、“俯仰”和和“偏航”,它们由操纵杆、油门和方向舵(粗略地说)机械地实现。为了编写模拟、跟踪和电子制导系统程序,绕三个轴的旋转可以表示为三个 3×3 旋转矩阵(图 4.2中给出的示例的 3-D 版本),这些矩阵组合起来即可得出飞行器所需的方向。例如,要调整偏航角和俯仰角,需要将两个相关矩阵相乘 - 如果计算出两个 3×3 矩阵相乘的所有行列步骤,则需要 27 次乘法和 18 次加法。但是,如果这些旋转用四元数而不是矩阵和滚转、俯仰、偏航角来表示,则只有 16 次乘法和 12 次加法。

For example, in modeling, simulating, tracking, or guiding an aircraft or spacecraft, its orientation (or “attitude”) is measured from three perpendicular axes—typically, along the length of the craft, across the width or wingspan, and the vertical up-and-down direction, with the origin at the centre of gravity. Rotations around these axes are called, respectively, “roll,” “pitch,” and “yaw,” and they are implemented mechanically by the yoke, throttle, and rudder (roughly speaking). To program simulations and tracking and electronic guidance systems, rotations about the three axes can be represented as three 3 × 3 rotation matrices—3-D versions of the example given in figure 4.2—and these are combined to give the required orientation of the craft. For instance, to adjust the yaw and pitch angles, you’d multiply the two relevant matrices—and if you count out all the row-by-column steps in multiplying two 3 × 3 matrices, you’ll find there twenty-seven multiplications and eighteen additions. But if these rotations are represented by quaternions instead of matrices and roll, pitch, yaw angles, it turns out that there are only sixteen multiplications and twelve additions.

但这还没有结束。与矩阵不同,四元数没有“万向节锁”问题,该问题曾困扰过阿波罗 11 号登月任务。万向节是一组环,传统上用于在运动过程中保持陀螺仪和其他仪器的稳定;一共有三个环,每个环围绕不同的轴旋转 - 但当其中两个对齐时,只有两个轴可用于定位飞行器。这在数学上反映为,当相关的旋转矩阵相乘时,最终会得到一个具有太多零的方向矩阵,因此它不再包含绕缺失轴旋转的数据。25因此,计算机会失去对第三维度方向的跟踪。四元数表示不会发生这种情况。

This is not the end of it, though. Unlike matrices, quaternions don’t have the problem of “gimbal lock,” which famously plagued the Apollo 11 mission to the moon. Gimbals are a set of rings that traditionally kept gyroscopes and other instruments steady during motion; there are three rings, each rotating around a different axis—but when two of them align, there are only two axes available for orienting the craft. This is reflected mathematically when the relevant rotation matrices are multiplied and you end up with an orientation matrix with too many zeroes, so that it can no longer include data for rotations about the missing axis.25 So, the computer loses track of the orientation in the third dimension. This doesn’t happen with quaternion representations.

然而,直到最近,四元数才在当今世界占据了一席之地。例如,NASA 于 1981 年首次将其常规用于编程制导、导航和控制系统中的旋转。他们的软件包括矢量、矩阵和四元数代数——因为它们都非常有用,不同的问题适合不同的方法。四元数在航天器定位和引导方面特别有用——例如,它们在模拟登月任务的轨道和跟踪 NASA 的火星探测器方面发挥了重要作用——并且在顺利处理从其收集的原始数据中生成的计算机图像方面也发挥了重要作用。

Yet it is only relatively recently that quaternions have found their place in today’s world. For instance, NASA first used them routinely to program rotations in guidance, navigation, and control systems in 1981. Their software includes vector, matrix, and quaternion algebra—for it is all marvelously useful, and different problems suit different approaches. Quaternions are especially useful both in orienting and guiding a spacecraft—for instance, they played an important role in simulating orbits for missions to the moon, and in tracking NASA’s Mars Exploration Rovers—and in smoothly processing computer images from the raw data it collects.

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除了这些技术应用之外,四元数旋转还有一些特殊之处——神秘而又令人惊奇。它们具有意想不到的性质,与描述我们世界基本构件(电子、质子和中子)的数学相同。即使你不想阅读图 4.4中旋转矢量计算的尾注,也请注意旋转矢量a表达式中的e i 2θ项。这很令人惊讶,因为它是通过将原始矢量p乘以单位四元数U = e i θ得出的。这是四元数旋转代数的一个怪癖,将 θ 变成 2θ。量子力学的一个怪癖是,虽然你可以将普通物质(例如球或行星)旋转 360° 并回到起点,但量子数学表明,电子等粒子需要“旋转” 720° 才能回到其原始状态。

In addition to all these technological applications, there’s something else that makes quaternion rotations special—something mysterious and utterly surprising. They have an unexpected property that they share with the maths describing the very building blocks of our world—electrons, protons, and neutrons. Even if you didn’t feel like reading the endnote on the calculations leading to the rotated vector in figure 4.4, notice in the caption the ei term in the expression for the rotated vector a. This is surprising, because it was found by multiplying the original vector p by the unit quaternion U = eiθ. It’s a quirk of quaternion rotation algebra that turns θ into 2θ. And it’s a quirk of quantum mechanics that although you can rotate an ordinary material thing, such as a ball or a planet, through 360° and get it back to its starting point, quantum maths says that a particle such as an electron needs to “spin” through 720° to get back to its original state.

电子并不是像地球或板球那样绕着自身旋转——根据大多数物理学家的说法,最好将电子想象成点,因此它们没有轴可以绕着旋转。但它们具有角动量,好像它们在旋转一样。这种奇异而出乎意料的现象是通过几年来巧妙地将两个和两个结合起来发现的,正是这种聪明的猜测和耐心的侦查工作让科学如此激动人心。所以,我将花一两页来追踪这一非凡发现的踪迹,因为它还将带我们更深入地了解量子力学和四元数之间迷人的联系。

Not that the electron is actually spinning about itself, like Earth or a cricket ball—according to most physicists, electrons are best imagined as points, so they have no axis to spin around. Yet they have an angular momentum as if they were spinning. This bizarre and unexpected phenomenon was discovered through some ingenious putting of two and two together over several years, the kind of brilliant guesses and patient detective work that makes science so thrilling. So, I’m going to take a page or two to follow the trail of this extraordinary discovery, for it will also take us deeper into the fascinating connection between quantum mechanics and quaternions.

发现“自旋”和四元数的联系

DISCOVERING “SPIN” AND THE QUATERNION CONNECTION

迈向“自旋”的第一步是不言而喻的事实,即电子可以具有轨道角动量,例如当它不是围绕自身旋转而是围绕原子核旋转时。但移动的电子会产生微小的磁场,正如丹麦教授汉斯·奥斯特德很久以前发现的那样。(1820 年,他注意到一个惊人的事实,即一股电流使附近的磁罗盘指针偏转,就好像电流既有磁性又有电性。这种装置很快成为这是第一种电报系统的基础,其中“指南针”指针指向表盘上的字母,不同的字母对应不同的电流量。)因此研究人员知道电子可以被其他磁场偏转,就像磁铁使铁屑偏转一样。通过这种方式偏转电子磁场,你可以了解很多关于电子磁场的特性——所以这就是奥托·斯特恩和沃尔特·格拉赫于 1922 年在汉堡着手做的事情。实际上,他们正在用一束蒸发的银原子做实验——电子对于这种装置来说偏转太多了——但银原子有一个与场相互作用的未配对电子。

The first step toward “spin” was the self-evident fact that an electron could have orbital angular momentum, such as when it rotates not around itself but around the nucleus of an atom. But moving electrons generate tiny magnetic fields, as the Danish professor Hans Øersted had discovered long ago. (In 1820 he’d noticed the astonishing fact that a burst of electric current deflected the needle of a nearby magnetic compass, as if the current were magnetic as well as electric. This kind of set-up quickly became the basis of the first telegraphy systems, where the “compass” needle pointed to letters on a dial, the different letters corresponding to different amounts of current.) So researchers knew that electrons could be deflected by other magnetic fields, just as a magnet deflects iron filings. You can tell a lot about the properties of an electron’s magnetic field by deflecting it in this way—so this is what Otto Stern and Walther Gerlach set out to do in Hamburg in 1922. Actually, they were experimenting with a beam of vaporised silver atoms—electrons would deflect too much for this set-up— but silver atoms have an unpaired electron that interacts with the field.

斯特恩的资历包括十年前成为爱因斯坦的第一位博士后学生,但现在他和格拉赫在吸引足够的实验资金方面遇到了麻烦——20 世纪 20 年代初和第一次世界大战之后,德国局势仍然动荡不安。然后他们的设备不断出现故障,直到经过一年的努力,他们才认为自己已经拍摄到了银原子偏转光束的足够好的图像,并将其捕捉到照相屏幕上。但他们并没有得到普通物理学所期望的结果,这表明蒸发原子的角动量的空间方向将是随机分布的,因为它们的方向在加热过程中会随机波动。事实上,他们一开始并没有得到什么结果。但当他们失望地盯着盘子时,两条黑线开始出现,就像某种“隐形墨水”魔法一样。显然,斯特恩抽的廉价雪茄含有大量的硫,当他检查盘子时,他呼出的硫与盘子上看不见的银原子混合,形成了黑色的硫化银!这两条黑线表明银原子角动量的分量只有两个可能的值。换句话说,角动量的空间取向是量子化的。

Stern’s credentials included having been Einstein’s first postdoctoral student a decade earlier, but now he and Gerlach were having trouble attracting sufficient funding for their experiment—in the early 1920s and the wake of World War I, things were still volatile in Germany. Then their equipment kept breaking down, until finally, after a year of struggle, they reckoned they’d taken a sufficiently good image of the deflected beam of silver atoms, which they’d captured on a photographic screen. But they didn’t get the results you’d expect from ordinary physics, which suggested that the spatial orientations of the vaporised atoms’ angular momenta would be randomly distributed, because of random fluctuations of their orientations during the heating process. In fact, they didn’t get much of a result at all, at first. But as they stared at the plate in disappointment, two black lines began to appear, as if by some kind of “invisible ink” magic. Apparently, Stern’s cheap brand of cigars had a high sulfur content, and as he was examining the plate, the sulfur in his breath mixed with the invisible silver atoms on the plate to produce black silver sulfide! These two black lines showed that there were just two possible values for the components of the silver atoms’ angular momenta. In other words, the spatial orientation of angular momentum was quantised.

他们并不完全惊讶于发现这样的结果——一个离散的而非连续的可能值范围——因为尼尔斯·玻尔的早期原子模型是量子化的。然而,随着量子力学的发展,玻尔模型的局限性逐渐显现出来,因此斯特恩和格拉赫到底发现了什么并不清楚。

They weren’t entirely surprised to find something like this—a discrete rather than continuous range of possible values—because Niels Bohr’s early model of the atom was quantised. As quantum mechanics developed, however, limitations in Bohr’s model came to light, so it wasn’t clear just what Stern and Gerlach had found.

随后,两位年轻的荷兰物理学家乔治·乌伦贝克和塞缪尔·古德斯米特提出,斯特恩和格拉赫测量的可能不是轨道角动量,而是自旋角动量——就好像产生磁场的电子围绕自身旋转一样。他们通过跟踪氢原子光谱中的一些异常现象得出了这个想法,氢原子每个原子只有一个电子。26乌伦贝克和古德斯米特利用磁场操纵电子的能级,发现如果电子的角动量仅取决于两个可能的独立值,那么得到的谱线将符合这一自旋假设——这正是斯特恩和格拉赫发现的。27

Then two young Dutch physicists, George Uhlenbeck and Samuel Goudsmit, suggested, in effect, that perhaps what Stern and Gerlach had measured were not orbital angular momenta but spin angular momenta—as if the electrons generating the magnetic fields were spinning around themselves. They’d come up with this idea by following up some anomalies in the atomic spectrum of hydrogen, whose atoms each have just one electron.26 Using magnetic fields to manipulate the energy levels of the electrons, Uhlenbeck and Goudsmit figured out that the resulting spectral lines would fit this spin hypothesis if the electron’s angular momentum depended on just two possible independent values—which is just what Stern and Gerlach had found.27

当乌伦贝克与荷兰传奇物理学家亨德里克·洛伦兹(Hendrik Lorentz)谈论这个奇怪的想法时(洛伦兹将在后面的相对论中提到),洛伦兹证实了他的担忧,他说,如果产生这种动量的电子真的旋转,那么它的速度一定高得不可思议。所以,电子并不是像板球一样旋转——它只是表现得像在旋转一样。这看起来太奇怪了,乌伦贝克绝望地拜访了他和古德斯米特的导师保罗·埃伦费斯特(Paul Ehrenfest),因为他们已经给他寄了一篇关于他们假设的论文。但埃伦费斯特很快就把它寄出去发表,告诉惊慌失措的乌伦贝克,“好吧,这是一个好主意,尽管它可能是错的。但你没有名气,所以你没什么可失去的。”这对年轻的乌伦贝克来说并没有多大安慰。28

When Uhlenbeck talked about this strange idea with the legendary Dutch physicist Hendrik Lorentz—who will come into this story later in connection with relativity—Lorentz confirmed his fears, saying that if the electron producing this momentum really were spinning, it would have to be doing so at an impossibly high speed. So, the electron isn’t spinning like a cricket ball—it just acts as if it is. Which seemed so bizarre that Uhlenbeck made a desperate visit to his and Goudsmit’s mentor, Paul Ehrenfest, for they’d already sent him a paper about their hypothesis. But Ehrenfest had promptly sent it off for publication, telling the panic-stricken Uhlenbeck, “Well, it’s a nice idea, though it may be wrong. But you don’t have a reputation, so you have nothing to lose.” Which wasn’t much comfort for young Uhlenbeck.28

当古怪的英国物理学家保罗·狄拉克开始发展电子行为的相对论量子力学描述时,一切都变得清晰起来。他发现,要使电子的波动方程具有相对论性,他必须加入一些量,这些量刚好给出乌伦贝克和古德斯米特发现的自旋动量值。德国物理学家沃尔夫冈·泡利已经开始研究自旋的非相对论数学,并引入了三个现在称为泡利矩阵的量,狄拉克在他的公式中采用了这些量。正如泡利明确指出的那样,这三个矩阵——有助于描述围绕三个空间轴的自旋角动量分量——以完全相同的方式相互关联作为汉密尔顿的单位四元数 i、j、k:将其中任意两个相乘,即可得到第三个的正负值——这与汉密尔顿八十五年前在布鲁姆桥上雕刻的规则相同!29

It all came together when the eccentric English physicist Paul Dirac began developing a relativistic quantum mechanical description of electron behavior. He found that to make an electron’s wave equation relativistic, he had to include quantities that gave just the values of spin momentum that Uhlenbeck and Goudsmit had found. The German physicist Wolfgang Pauli had already made a start on the nonrelativistic mathematics of spin and had introduced three quantities now known as Pauli matrices, which Dirac adapted in his formulation. As Pauli spelled out explicitly, these three matrices—which help describe spin angular momentum components about the three spatial axes—relate to each other in exactly the same way as Hamilton’s unit quaternions i, j, k: multiply any two of them and you get plus or minus the third—the same rules Hamilton had carved on Broome Bridge eighty-five years earlier!29

此外,数学表明,如果你想旋转量子粒子的自旋轴,旋转方程类似于四元数旋转的方程。这就是为什么电子——以及物质的其他构成要素——具有这种奇怪的特性,你需要将它们旋转两次才能回到起点。30

What’s more, the maths says that if you want to rotate the spin axis of a quantum particle, the rotation equations are analogous to those for quaternion rotations. That’s why electrons—and the other building blocks of matter—have that strange property where you need to rotate them twice to get back to where you started.30

不过,正如我所强调的那样,亚原子粒子实际上并不像板球那样旋转;相反,它们的自旋与磁场对齐,因此如果你能慢慢旋转磁场,你就会旋转自旋轴(如果不是粒子本身的话)。这正是实验物理学家在这一惊人数学结果出现半个世纪后成功做到的。1975 年,两位澳大利亚物理学家托尼·克莱因和杰夫·奥帕特进行了一项实验,表明这种奇怪的旋转行为是物理现象——而不仅仅是数学现象。他们的实验与托马斯·杨著名的双缝实验有相似之处,后者表明两束光之间的干涉图案是波的干涉图案,而不是粒子的干涉图案。克莱因和奥帕特将一束中子(以物质波的形式传播)衍射成两部分;然而,在这种情况下,一部分与旋转自旋轴的外部磁场相互作用,另一部分保持不变。这两束光产生的干涉图样表明,自旋必须旋转两个周期才能回到原来的状态。这正是数学所预测的!31

As I’ve emphasised, though, subatomic particles don’t really rotate like cricket balls; rather their spin aligns with a magnetic field, so if you could slowly rotate the field, you’d be rotating the spin axis (if not the particle itself ). This is just what experimental physicists succeeded in doing, half a century after this astounding mathematical result. In 1975, two Australian physicists, Tony Klein and Geoff Opat, carried out an experiment that showed that this strange rotational behavior was physical—it wasn’t just a mathematical artifact. Their experiment has some kinship with Thomas Young’s famous two-slit experiment, which showed that the interference pattern between two beams of light was that of waves, not particles. Klein and Opat diffracted a beam of neutrons (traveling as a matter wave) into two parts; in this case, however, one part interacted with an external magnetic field that rotated the spin axis, and the other remained unchanged. The interference pattern made by these two beams showed that the spin had to be rotated through two cycles to get back to its original state. Which is just what the maths predicted!31

这是一个令人震惊的结果,但在克莱因和奥帕特将他们的研究成果发表之前,另外两个团队也证实了这个奇怪的数学预测。32自旋,无论它是什么,绝对是真实存在的,它的行为遵循汉密尔顿在最终设法在三维空间中旋转矢量时发现的相同旋转数学怪癖。

It was a stunning result, but before Klein and Opat could get their work into print, two other teams also confirmed this strange mathematical prediction.32 Spin, whatever it was, was definitely real, and it behaved according to the same quirk of rotational maths that Hamilton found when he finally managed to rotate vectors in three-dimensional space.

如今,“旋转”已成为日常生活中不可或缺的一部分,例如用于医学诊断的磁共振成像 (MRI)。MRI 可拍摄患者内脏器官的 3D 图像,因此无需进行侵入性诊断手术或放射治疗,它通过使用磁场来调整氢原子的自旋,由于我们体内水分和脂肪含量高,氢原子无处不在。然后,这些自旋通过无线电波发生偏转,不同的无线电脉冲序列突显身体的不同部位;当无线电源关闭时,图像就会被捕获,自旋会恢复到平衡状态,并在这一过程中释放出电磁能。

Today “spin” is a vital part of everyday life—in the magnetic resonance imaging (MRI) used for medical diagnoses, for example. MRI takes 3-D images of a patient’s internal organs—so there’s no need for invasive diagnostic surgery or radiation—and it does this by using magnetic fields to align the spins of hydrogen atoms, which are ubiquitous given the amount of water and fat in our bodies. The spins are then deflected via radio waves, different sequences of radio pulses highlighting different parts of the body; images are captured when the radio source is turned off and the spins return to their equilibrium state, giving off electromagnetic energy in the process.

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如果汉密尔顿知道四元数与电子自旋这一基本特性有关,他一定会欣喜若狂。如果他能看到四元数为地球带来的宇宙的壮丽景象,他那充满诗意的灵魂就会为这一切的奇妙、宏伟和神秘而激动不已——他曾在自己最喜欢的一首诗《月全食颂》中试图捕捉这种神秘。这首诗写于他还不到 18 岁,一直保存到去世。在诗中,他向月亮女神询问月食期间“深红色”笼罩在她“美丽的额头”上的“不祥的阴暗”,问这是由来自另一个世界的云朵还是巫师的魔法造成的?或者,也许是我们地球的“阴影锥”使月亮在“我们好奇的眼睛”眼中变得昏暗?这种诗意的想象与科学的结合是汉密尔顿的典型风格。他很想知道,他的四元数(以及现在被称为哈密顿动力学,量子力学就是以此为基础建立的)是否有助于揭示宇宙中一些最神奇的秘密。33

Hamilton would be beside himself with joy if he knew that quaternions are related to something as fundamental as this strange electron spin property. And if he could see the glorious pictures of our universe that quaternions have helped bring to Earth, his poetic soul would thrill to the wonder and grandeur and mystery of it all—the kind of mystery he’d tried to capture in one of his favourite poems, Ode to the Moon under Total Eclipse. He’d written it when he was not quite eighteen, and he kept it till the day he died. In the poem, he questions the moon goddess on the “portentous gloom” as “crimson red” spreads over her “lovely brow” during the eclipse, asking if it be caused by some passing cloud from another world, or a wizard’s magic spell? Or, perhaps, is it our Earth’s “shadowy cone” that dims the moon to “our wondering eye?” This mix of poetic imagination and science is typical of Hamilton. He would have loved to know that his quaternions (and also what is now known as Hamiltonian dynamics, on which quantum mechanics was modeled) have helped bring to light some of the universe’s most magical secrets.33

然而,当他第一次发现四元数时,汉密尔顿只瞥见了它们更广泛的可能性。然而,从一开始,他就深信四元数在物理应用中起着至关重要的作用——尽管在他的一生中,四元数从未完全实现他的期望,从未像今天与 NASA 和量子力学的联系那样引人注目,但他仍然拥有自己的荣耀。例如,1848 年,他因发现四元数而获得了爱尔兰皇家学院和爱丁堡皇家学会的奖章。

When he first discovered quaternions, though, Hamilton only glimpsed their wider possibilities. Yet right from the beginning he felt in his bones that quaternions had a vital role to play in physical applications—and if in his lifetime they never quite lived up to his hopes, never achieving the glamorous profile that they have today by association with NASA and quantum mechanics, still he had his share of glory. In 1848, for example, he was awarded medals for his discovery of quaternions, from both the Royal Irish Academy and the Royal Society of Edinburgh.

他长期致力于理解复数,将其扩展到三维,并发明了四维四元数,这让我们想起了即使是最简单的事情,我们今天也认为理所当然,但理解这些事情需要一种深刻的思考。德摩根和皮科克在符号代数方面的工作也是如此,他们梳理出了人们长期以来一直在使用的代数定律,却从未意识到它们可能适用于数字以外的事物。今天,主要是数学哲学家讨论数字与符号、语言和意义之间的关系。教育政策制定者和资助机构主要关注数学如何发挥作用。因此,值得再次强调的是,汉密尔顿不知道他的四元数在驾驶机器人和航天器等高科技应用中会有多么有用。他也没有预见到向量将在物理、工程和 IT 中发挥多么普遍的作用,因为现在它们几乎出现在你需要指定物体空间位置或存储和处理数据的任何地方。这表明,尽管现代数学源于古代试图解决美索不达米亚人面临的实际问题,尽管解决实际问题的需要仍然是新数学见解的主要驱动力,但有时,为数学而数学却能解决未来的技术和物理问题。

His long struggle to make sense of complex numbers, to expand them to three dimensions and to invent his 4-D quaternions, reminds us of the kind of deep thinking needed to make sense of even the simplest things we take for granted today. The same goes for De Morgan and Peacock’s work on symbolic algebra, teasing out the laws of algebra that people had long been using without ever realising they might apply to things other than just numbers. Today, it is mainly philosophers of mathematics who discuss such things as the relationship between numbers and symbolism, language, and meaning. Educational policymakers and funding bodies are focussed mainly on how mathematics can be useful. So it is worth saying again that Hamilton had no idea just how useful his quaternions would prove in such high-tech applications as driving robots and spacecraft. Nor did he foresee what a ubiquitous role vectors would play in physics, engineering, and IT, for now they crop up just about everywhere you need to specify the spatial positions of objects or to store and handle data. Which goes to show that although modern maths has its ancient roots in attempts to solve the kinds of practical problems the Mesopotamians were faced with—and although the need to solve practical problems is still a key driver of new mathematical insights—sometimes it is maths for maths’s sake that leads to solutions of future technological and physical problems.

事实上,在汉密尔顿告诉他的朋友约翰·格雷夫斯他的四元数的几个月后,格雷夫斯就发明了“八元数”——不是四维的,而是由四元数对组成的八维复数。它们只是一种好奇,证明代数世界除了数字和四元数之外还有更多的东西——然而今天人们正在研究它们与粒子物理标准模型的可能联系。这是当前的模型,其中物理现实的构成要素被分为具有相似属性的类别。例如,物质由不同种类的夸克和轻子组成,后者是带电的粒子,比如电子。还有力的载体或玻色子,比如携带电磁力的光子,以及携带质量的希格斯玻色子。

In fact, a couple of months after Hamilton told his friend John Graves about his quaternions, Graves invented “octonions”—not 4-D but 8-D complex numbers made from pairs of quaternions. They were just a curiosity, proof that there was even more to the algebraic world than numbers and quaternions—yet today they are being investigated for possible connections with the standard model of particle physics. This is the current model in which the building blocks of physical reality are classified in categories that share similar properties. For instance, matter is made of different kinds of quarks and leptons, the latter being particles with charge, such as electrons. Then there are the force-carriers or bosons, such as the photons that carry electromagnetic force, plus the Higgs boson, which carries mass.

目前还不清楚八元数是否会在这种背景下发挥作用——但正如汉密尔顿非交换乘法的应用一样,一旦一个想法被提出,你就永远不会不知道它会激发什么。34乔治·布尔是另一个例子;他的代数逻辑系统是从皮科克和德·摩根奠定的符号代数基础以及汉密尔顿的四元数发展而来的。布尔在一场倾盆大雨中感冒后突然去世,当时他还太年轻,但他并不知道,他用 0 和 1 来表达真假陈述的系统最终会驱动我们的计算机和其他电子设备和小工具。

The jury is still out on whether or not octonions will prove useful in this context—but as with the applications that flowed from Hamilton’s noncommutative multiplication, once an idea is out there, you never know what it may inspire.34 George Boole is another example; his system of algebraic logic evolved from the foundations of symbolic algebra laid by Peacock and De Morgan, and by Hamilton’s quaternions. Little did Boole know, when he died suddenly and far too young after catching cold in a downpour of rain, that his system of expressing true and false statements in terms of 0s and 1s would eventually drive our computers and other electronic devices and gadgets.

然而,早在 19 世纪 40 年代,矢量分析的发展还有很长的路要走,它不仅拓展了我们的技术,还拓展​​了我们对数学和宇宙的理解。在接下来的几章中,我们将遇到新一代数学家和数学物理学家,他们开发汉密尔顿矢量和四元数的工作将带我们进入更加奇妙的现实领域。

Back in the 1840s, though, there was still a way to go in developing the vector analysis that has broadened not just our technology but also our understanding of maths, and of our universe. Over the next few chapters, we’ll meet new generations of mathematicians and mathematical physicists, whose work developing Hamilton’s vectors and quaternions will take us into yet more wondrous realms of reality.

不过,首先,我们会看到,新想法总是需要时间来证明其价值。我们还会遇到一位非凡的局外人,他发明了自己的强大矢量系统。他的动机与汉密尔顿不同——他对旋转和复数不感兴趣——但最终,我们会看到,这两种方法都有助于数学家开发当今如此重要的复杂矢量和张量分析。

First, though, we’ll see that it always takes time for new ideas to prove their worth. And we’ll meet an extraordinary outsider, who invented his own powerful vectorial system. His motivation was different from Hamilton’s—he wasn’t interested in rotations and complex numbers—but ultimately, we’ll see that both approaches would help mathematicians to develop the sophisticated vector and tensor analyses that are so important today.

(5)令人意外的新球员和缓慢的接受度

(5) A SURPRISING NEW PLAYER AND A VERY SLOW RECEPTION

如今我们知道汉密尔顿的四元数是多么有用,但在当时,它们面临着一场激烈的争论。1847 年,汉密尔顿向他的朋友罗伯特·格雷夫斯幽默地讲述了他在英国科学促进会会议上发表关于四元数的论文后讨论的情况。顺便说一句,正是在 1833 年的一次英国协会会议上,“科学家”一词首次被引入,目的是将科学和数学研究人员置于与艺术家相同的专业地位。这个想法是威廉·惠威尔提出的,他还在 1840 年创造了“物理学家”这个名字。无论如何,1847 年,汉密尔顿告诉格雷夫斯,虽然皮科克博士认为他已经“充分阐述”了他的四元数系统,但另一位听众却怀疑“使用我的新微积分时可能会犯错误。作为回应,我否认有权对犯错的能力设置任何限制。” 1

These days we know how useful Hamilton’s quaternions are, but at the time they faced quite a battle for acceptance. In 1847, Hamilton gave his friend Robert Graves a humorous account of the discussion that followed his presentation of a paper on the subject, at a meeting of the British Association for the Advancement of Science. Incidentally, it was at a British Association meeting in 1833 that the term “scientist” was first introduced, in order to put scientific and mathematical researchers on the same professional footing as artists. The idea was William Whewell’s, and he also coined the name “physicist” in 1840. Anyway, in 1847 Hamilton told Graves that while Dr. Peacock thought he’d given “a capital exposition” of his quaternion system, another member of the audience wondered about “the possibility of making mistakes in the use of my new calculus. In reply to which I disclaimed the power of setting any limit to the faculty of making blunders.”1

汉密尔顿继续说,这种“温和的小冲突”还有几次,然后博学家约翰·赫歇尔爵士给四元数带来了狂热的认可。这对皇家天文学家乔治·艾里来说太过分了。艾里曾称汉密尔顿对圆锥折射的数学预测(我们在第 2 章中看到)“可能是有史以来最了不起的预测”——但他对四元数代数的看法要悲观得多。汉密尔顿告诉格雷夫斯,赫歇尔发言后,艾里站起来“讲述他自己对 [四元数] 的了解,他承认自己根本不了解四元数;但 [他] 让我们明白,他不知道的东西不值得知道。”换句话说,汉密尔顿讽刺地补充道,艾里似乎认为任何在看来晦涩难懂或自相矛盾的东西一定是错误的。2

There were a few more “gentle skirmishes” of this sort, continued Hamilton, and then the polymath Sir John Herschel gave quaternions a rapturous endorsement. Which was too much for the Astronomer Royal, George Airy. Airy had called Hamilton’s mathematical prediction of conical refraction (which we saw in chap. 2) “perhaps the most remarkable prediction that has ever been made”—but he took a much dimmer view of quaternion algebra. Hamilton told Graves that after Herschel spoke, Airy rose to his feet “to speak of his own acquaintance with [quaternions], which he avowed to be none at all; but [he] gave us to understand that what he did not know could not be worth knowing.” In other words, Hamilton added wryly, Airy seemed to assume that anything that to him seemed obscure or paradoxical must be erroneous.2

如果像汉密尔顿这样地位的科学家都难以让自己的系统被接受,想象一下,对于一个生活在偏远乡村小镇的自学成才的孤独者来说,这将会有多困难。空气中一定有某种矢量,因为这样的人确实存在:德国教师赫尔曼·格拉斯曼。在科学界经常发生的独立平行发现中,格拉斯曼发现了一个与汉密尔顿的系统类似的系统,这是其中的一个非凡例子。其他几个人——尤其是因“莫比乌斯带”而出名的德国人奥古斯特·莫比乌斯和意大利人朱斯托·贝拉维蒂斯——也一直在向这个想法靠拢。但最终成功的是汉密尔顿和格拉斯曼——两人都不知道对方是谁。

If someone of Hamilton’s scientific standing was having trouble getting his system accepted, imagine how much harder it would be for a self-taught loner living in a remote country town. There must have been something vectorial in the air, for such a person actually existed: the German schoolteacher Hermann Grassmann. In one of those remarkable instances of independent parallel discovery that happen surprisingly often in science, Grassmann discovered a system analogous to Hamilton’s at the very same time. Several others, too—notably the German August Möbius, of “Möbius strip” fame, and the Italian Giusto Bellavitis—had also been inching toward the idea. But it was Hamilton and Grassmann who ultimately succeeded—each unbeknownst to the other.

格拉斯曼比汉密尔顿小四岁,他的父亲是波美拉尼亚神学家尤斯图斯·格拉斯曼,他在斯德丁(现称什切钦,因为该地区位于今天的波兰西北部)的当地高中教授数学和科学。赫尔曼和他的十一个兄弟姐妹都出生在斯德丁,但在学校里,他没有表现出任何数学天赋,不像神童汉密尔顿——尤斯图斯认为他的儿子可能会找到一份园丁或工匠的工作。尽管如此,赫尔曼还是设法考上了柏林大学,在那里学习了语言学和神学——不管他父亲怎么想,他对成为一名园丁并不感兴趣。事实上,他的目标是成为一名像尤斯图斯一样的教师,所以当他从大学回家后,他开始自学数学和科学。他的父亲编写了一些数学教科书,显然这些教科书效果很好,因为赫尔曼很快就通过了国家教师考试。3

Four years younger than Hamilton, Grassmann was the son of a Pomeranian theologian, Justus Grassmann, who taught maths and science at the local high school in Stettin—now spelled Szcezcin, because the area is in today’s northwest Poland. Hermann and his eleven siblings were born in Stettin, but at school he’d shown no mathematical talent at all, unlike the prodigy Hamilton—Justus thought his son might find a job as a gardener or craftsman. Still, Hermann managed to get into Berlin University, where he studied philology and theology—he wasn’t interested in becoming a gardener, no matter what his father thought. In fact, he’d set his eyes on becoming a teacher like Justus, so when he returned home from university, he began to study mathematics and science on his own. His father had written some maths textbooks, and they evidently worked well, for soon Hermann passed the state teaching exams.3

在柏林教了一年书之后,他便在家乡教授高中数学、科学、语言和神学,并度过了余生。与同样自学数学,后来当了多年教师的乔治·布尔不同,赫尔曼·格拉斯曼从未获得同行的认可,因此无法获得大学职位。这当然不是因为缺乏才华:格拉斯曼在学校可能没有表现出多少天赋,但他是一位才华横溢的独创数学家。这也不是因为他缺乏努力——他非常有野心,并付出了非凡的努力来争取职业晋升。

After a year teaching in Berlin, he taught high school maths, science, languages, and theology in his hometown for the rest of his life. Unlike George Boole, who also taught himself mathematics and then spent years as a schoolteacher, Hermann Grassmann never gained the recognition from his peers that might have led to a university position. It certainly wasn’t for lack of merit: Grassmann may not have shown much talent at school, but he was a brilliantly original mathematician. And it wasn’t for want of trying, either—he was quite ambitious, and put in extraordinary efforts to try to gain professional promotions.

他开始通过深入研究潮汐来证明自己,并在此过程中意识到,就像牛顿一样,空间中的物理现象是矢量的(他们俩都没有使用这个术语)。但是,虽然牛顿只使用了平行四边形规则以及力和速度既有大小又有方向的思想,但格拉斯曼却像汉密尔顿一样着迷于发展矢量代数的底层数学的挑战。当他将这种代数应用于潮汐理论时,他发现自己“对这种方法得出的计算结果的简单性感到震惊” 。4

He began trying to prove himself with an intensive study of the tides, and in the process he realised, like Newton, that physical phenomena acting in space are vectorial (not that either of them used that term). But while Newton used just the parallelogram rule and the idea that forces and velocities have both magnitude and direction, Grassmann became as entranced as Hamilton by the challenge of developing the underlying mathematics of vector algebra. And when he applied this algebra to the theory of tides, he found himself “astounded by the simplicity of the calculations resulting from this method.”4

1840 年,格拉斯曼准备将自己长达 200 页的潮汐理论论文提交给决定教师晋升的柏林考试委员会。不幸的是,但也许并不奇怪,考官未能认识到论文革命性的数学基础。格拉斯曼一定有非凡的自信,独自工作,无人认可他的成就。但他坚信新方法能够简化计算,因此,尽管关于潮汐的论文被拒,以及因此无法晋升,他似乎并没有气馁,在 1844 年,他发表了一篇关于新方法的巨作。他把它命名为Die Lineale Ausdehnungslehre,即“线性扩展理论”。

By 1840, he was ready to submit his 200-page dissertation on tidal theory to the Berlin examination committee that decided on teacher promotions. Unfortunately, but perhaps not surprisingly, the examiner failed to appreciate its revolutionary mathematical basis. Grassmann must have had extraordinary self-belief, working away in isolation, with no one to recognise his achievements. But he passionately believed in the power of his new method to simplify computations, and so, seemingly undaunted by the rejection of his paper on the tides—and by the consequent lack of promotion up the teaching ladder—in 1844 he published a monumental treatise on his new approach. He called it Die Lineale Ausdehnungslehre, or “The Linear Theory of Extensions.”

格拉斯曼令人惊叹的AUSDEHNUNGSLEHRE

GRASSMANN’S AMAZING AUSDEHNUNGSLEHRE

汉密尔顿谈到矢量,而格拉斯曼则使用了德语单词strecke,可以翻译为“距离、路线、线或延伸”——他的基本几何对象确实是“线”,可以“拉伸”或“扩展”形成平面,就像点可以延伸成线一样。这听起来并不那么惊天动地,但其中激进而巧妙的部分是格拉斯曼构建几何线及其扩展的代数的方式——就像汉密尔顿一直在寻找四元数和三维旋转的代数一样。汉密尔顿惊讶地发现四元数和向量的乘法并不总是可交换的,格拉斯曼报告说他

While Hamilton spoke of vectors, Grassmann used the German word strecke, which can be translated as “distance, route, line, or stretch”—and his basic geometric objects were indeed “lines,” which could be “stretched” or “extended” to form planes, just as points could be extended into lines. This doesn’t sound all that earth-shattering, but the radical and ingenious part of it was the way Grassmann built an algebra of geometrical lines and their extensions—just as Hamilton had been searching for an algebra of quaternions and 3-D rotations. And while Hamilton had been shocked to discover that multiplication of quaternions and vectors was not always commutative, so Grassmann reported that he was

最初对这个奇怪的结果感到困惑:虽然普通乘法的其他定律(包括乘法与加法的关系)成立,但只有同时改变符号(即把 + 改为 − ,把 − 改为 + )才能交换因子。5

initially perplexed by the strange result that though the other laws of ordinary multiplication (including the relation of multiplication to addition) held, yet one could only exchange factors if one simultaneously changed the sign (i.e. changed + to − and − to +).5

这是一种冗长的说法,表明他的几何“线”的乘法不交换——这表明皮科克和德·摩根在识别和命名算术基本定律方面的工作在概念和语言上具有重要意义。事实上,格拉斯曼关于将 + 改为 − 的说法,是一种表明向量积是交换的。这正是叉积的右手定则所显示的(正如我们在图 4.1中看到的):如果你将手指从p弯曲到q,拇指指向上方,那么如果你将它们从另一个方向弯曲到p,你的拇指就会指向下方——这意味着p × q = − q × p

This was a long-winded way of saying that multiplication of his geometric “lines” was not commutative—which goes to show the conceptual and linguistic importance of Peacock and De Morgan’s work on identifying and naming the fundamental laws of arithmetic. In fact, Grassmann’s talk of changing + to − was a way of saying that vector products are anticommutative. This is just what the right-hand rule for cross products shows (as we saw in fig. 4.1): if you curl your fingers from p to q and your thumb points upward, then if you curl them the other way, from q to p, your thumb will point downward—which means that p × q = −q × p.

我之前提到过,年轻的汉密尔顿研究过拉普拉斯的《天体力学》,发现平行四边形规则的逻辑存在错误,而他之所以发现四元数,是因为他对朋友德摩根的符号代数工作和沃伦的复数几何论文很感兴趣。格拉斯曼也知道平行四边形规则,但除此之外,他通过一条非常不同的途径得出了他的“扩展”理论。

I mentioned earlier that the young Hamilton had studied Laplace’s Mécanique Céleste and found an error in the logic of the parallelogram rule, and that he had discovered quaternions because he’d been intrigued by his friend De Morgan’s work on symbolic algebra and by Warren’s paper on the geometry of complex numbers. Grassmann also knew about the parallelogram rule, but otherwise he came to his theory of “extensions” by a very different route.

他从意大利裔法国数学家约瑟夫-路易·拉格朗日的一本教科书中了解到代数和微积分在物理学中的力量。这本书名为分析力学》,它曾被拉普拉斯的《天体力学》问世于半个世纪前——1788 年,即法国大革命前夕;而拉普拉斯的《天体力学》问世则晚了十年。拉格朗日和拉普拉斯对政治并不特别感兴趣,在革命年代里设法保持中立。1790 年,法国科学院成立了度量衡委员会——拉普拉斯和拉格朗日都是委员会成员——但当 1793 年恐怖统治开始时,狂热的新政府关闭了科学院。他们还算明智,让度量衡委员会继续运作——除了一些他们认为可疑的成员,包括外国人6——正是由于这个委员会,公制才得以创立。

He learned about the power of algebra and calculus in physics from a textbook by the Italian-French mathematician Joseph-Louis Lagrange. It was called Mécanique Analytique (Analytical Mechanics), and it had been published half a century earlier—in 1788, the eve of the French Revolution; Laplace’s Celestial Mechanics appeared a decade later. Lagrange and Laplace were not particularly interested in politics and managed to steer a neutral course through the Revolutionary years. In 1790 the French Academy of Science created a Committee on Weights and Measures—both Laplace and Lagrange were members—but when the Reign of Terror began in 1793, the fanatical new government closed the Academy. They did have the sense to keep the weights and measures committee going—minus a few members they deemed suspect, including foreigners6—and it is thanks to this committee that the metric system was created.

至于《分析力学》,拉普拉斯更新了牛顿关于行星运动的著作——格拉斯曼以拉普拉斯为基础提出了潮汐理论——而拉格朗日则扩展了牛顿的一般运动概念。我提到约翰·伯努利、埃米莉·杜·夏特莱,尤其是莱昂哈德·欧拉已经开始将牛顿的著作翻译成莱布尼茨微积分的语言,但拉格朗日扩展了微分方程的理论,使它们最终成为力学的语言——即我们今天习以为常的运动和力的语言。拉格朗日是父母十一个孩子中唯一一个活过童年的人,他确实让他们感到骄傲。7

As for Mécanique Analytique, while Laplace had updated Newton’s work on planetary motion—Grassmann built on Laplace for his theory of tides—Lagrange extended Newton’s concept of motion in general. I mentioned that Johann Bernoulli, Émilie du Châtelet, and especially Leonhard Euler had begun translating Newton’s work into the language of Leibnizian calculus, but Lagrange extended the theory of differential equations so that they finally became the language of mechanics—the language of motion and forces that we take for granted today. Lagrange was the only one of his parents’ eleven children to survive beyond childhood, and he certainly did them proud.7

格拉斯曼在《解析学》的前言中承认自己受益于《分析力学》 ——但他指出,使用自己的新系统,“计算结果往往比拉格朗日的工作短 10 倍以上。”他指的是,如果使用四元数进行复合旋转,表达式的简洁性也类似。然而,格拉斯曼的向量之旅并非始于拉格朗日,而是始于他父亲的教科书——拥有这样的父亲是他长期努力争取数学成功的少数幸运之一。

Grassmann acknowledged his debt to Mécanique Analytique in the foreword of Ausdehnungslehre—but he noted that in using his own new system, “the calculations often came out more than ten times shorter than in Lagrange’s work.” What he meant was a similar economy of expression as you get if you use quaternions for compound rotations. Grassmann’s journey to vectors had begun not with Lagrange, however, but with his father’s textbooks—and having such a father was one of the few pieces of luck he had in his long struggle for mathematical success.

正如他在《几何学》的前言中指出的那样,他的父亲将矩形定义为其长度和宽度的“几何乘积”。这本身就很新颖——你可以将两个几何量相乘创建一个新的几何对象——在这种情况下,“乘以”两条线得到一个矩形。相比之下,这种传统的乘积上下文将表示长度和宽度的数字相乘得到另一个数字,古人很久以前就将其定义为矩形的面积。但赫尔曼将他父亲的想法带入了真正的矢量领域,他指出,如果你把边“不仅仅是长度,而是有向量”的话,你可以对所有平行四边形做同样的事情。当然,“有向量”是汉密尔顿“矢量”的另一个名字。

As he noted in the foreword to Ausdehnungslehre, his father had defined a rectangle as the “geometrical product” of its length and width. This was novel in itself—the idea that you could multiply two geometric quantities in themselves to create a new geometric object—in this case, “multiplying” two lines to get a rectangle. By contrast, the traditional product in this context multiplies the numbers representing the length and width to get another number, which the ancients had long ago defined as the area of the rectangle. But Hermann took his father’s idea into true vectorial territory by noting that you could do the same for all parallelograms, if you viewed the sides “not merely as lengths, but rather as directed magnitudes.” And, of course, “directed magnitude” is another name for Hamilton’s “vector.”

您可能还记得,向量积a × b的模的几何定义是 | a || b |sin θ,其中 | a | 表示向量a的模,| b |也是同样如此。这就是为什么还有另一个关于叉积的笑话(来自电视情景喜剧《班长》):“将大象和葡萄放在一起会得到什么?”答案是:“大象 葡萄 sine-theta!”更严肃地说,对于矩形,θ = 90°,因此 sin θ = 1,这意味着您又回到了传统规则,即只需将边的模相乘即可找到面积。换句话说,sin θ 的作用对于矩形是隐藏的,但对于其他平行四边形则不是。

You might remember from school that the geometric definition of the magnitude of the vector product a × b is |a||b|sin θ, where |a| means the magnitude of the vector a, and similarly for |b|. That’s why there’s yet another cross-product joke (from the television sitcom Head of the Class): “What do you get when you cross an elephant and a grape?” Answer: “Elephant grape sine-theta!” More seriously, in the case of a rectangle, θ = 90°, and so sin θ = 1, which means you’re back to the traditional rule where you find the area simply by multiplying the magnitudes of the sides. In other words, the role of sin θ is hidden for rectangles, but not for other parallelograms.

格拉斯曼将这种几何乘法称为“外积”。在三维空间中,它的解释与汉密尔顿(和现代)的矢量积略有不同,如您在尾注中看到的那样,但在计算上是相同的。8格拉斯曼还定义了我们(遵循汉密尔顿)所说的两个矢量的“标量积”,但他称之为“内积”。他使用您可能已经学过的标量积的相同几何定义来定义他的内积,ab = | a || b |cos θ——尽管他当然没有这种现代矢量符号。

Grassmann called this kind of geometrical multiplication an “outer product.” In three dimensions, it differs slightly in its interpretation from Hamilton’s (and the modern) vector product, as you can see in the endnote, but computationally it is the same.8 Grassmann also defined what we, following Hamilton, call the “scalar product” of two vectors, but which he called an “inner product.” He defined his inner product using just the same geometrical definition you may have learned for the scalar product, ab = |a||b|cos θ—although he didn’t have this modern vectorial notation, of course.

虽然我们今天在向量分析中使用汉密尔顿的术语“向量”和“标量”,但我们稍后会看到,格拉斯曼的“内”和“外”积最终也会找到现代的归宿,分别作为n维向量和张量的乘积的名称。汉密尔顿曾告诉德摩根他认为没有理由找不到他称之为“多元组”(与“三元组”类似)的代数,形式为 ( a1 a2 …、an ),但他止步于四元数,其中n = 4(或当重点放在向量部分时为 3)。格拉斯曼的方法也侧重于三维,但他以一种抽象的方式呈现了它,这种方式很容易适应任意维数——事实上,他应用它开创了n维线性代数的新理论。

While we use Hamilton’s terms “vector” and “scalar” in vector analysis today, we’ll see later that Grassmann’s “inner” and “outer” products would also eventually find a modern home, as the names for products of n-dimensional vectors and tensors, respectively. Hamilton had told De Morgan he saw no reason you couldn’t find an algebra of what he called “polyplets” (by analogy with “triplets”) of the form (a1, a2, … , an), but he had stopped with quaternions, where n = 4 (or 3 when the focus is on the vector part). Grassmann’s approach also focussed on three dimensions, but he presented it in an abstract way that was readily adaptable to any number of dimensions—in fact, he applied it to pioneer the new theory of n-dimensional linear algebra.

格拉斯曼的代数遵循除外积交换律之外的所有常规定律——和汉密尔顿一样,他知道为他的新代数系统建立这些规则很重要。但它比汉密尔顿的系统更进一步,因为它不仅包含向量,还包含张量代数的萌芽。不过,在这个故事的这个阶段,必须说格拉斯曼自学的数学风格相当令人费解。

Grassmann’s algebra obeyed all the usual laws except the commutative law for outer products—like Hamilton, he knew it was important to establish these rules for his new algebraic system. But it went further than Hamilton’s system in that it contained not just vectors but the germ of tensor algebra. At this point in the story, though, it must be said that Grassmann’s self-taught mathematical style was rather impenetrable.

汉密尔顿在 1853 年出版的 800 页的《四元数讲座》中并没有更清晰地表达。首先,他和格拉斯曼都试图表达他们的同行很难理解的全新思想 — — 矢量分析、线性代数的基本原理等等:汉密尔顿是三维复数和旋转,格拉斯曼是n维矢量空间和原张量代数。他们也都开始发展矢量微积分(下一章会详细介绍)。但是他们工作的原创性意味着他们每个人都必须发明新词来描述他们所做的事情 — — 与最终流行起来的少数几个词(尤其是我刚才提到的“矢量”、“标量”、“内积”和“外积”)相比,这些词更加晦涩难懂,而且最终都是多余的。两人都热衷于哲学思考——汉密尔顿的时间哲学和格拉斯曼对数学基本概念的哲学方法。例如,格拉斯曼是辩证法的忠实拥护者,他运用辩证法将概念分为“相等-不同”、“真实-形式”等类别。

Hamilton’s wasn’t much clearer in his expanded 800-page Lectures on Quaternions, which he published in 1853. For a start, there was the fact that both he and Grassmann were trying to articulate brand-new ideas that their peers would find difficult to digest anyway—the fundamentals of vector analysis, linear algebra, and more: in Hamilton’s case, three-dimensional complex numbers and rotations, and in Grassmann’s, n-dimensional vector spaces and prototensor algebra. They both began to develop vector calculus, too (of which more in the next chapter). But the originality of their work meant that each of them had to invent new words to describe what they were doing—dozens more opaque and ultimately superfluous words than the few that finally caught on (notably the “vector,” “scalar,” “inner,” and “outer” products that I just mentioned). Both men also had a penchant for an excess of philosophising—Hamilton’s philosophy of time, and Grassmann’s philosophical approach to the fundamental concepts of mathematics. For instance, Grassmann was a devotee of dialectics, and he applied it in categorising concepts as “equal-different,” “real-formal,” and so on.

汉密尔顿已经享有数学天才的声誉,而格拉斯曼只是一位默默无闻的乡村教师,这很有帮助——他居然能为《演讲》找到出版商,这真是太神奇了。就连汉密尔顿在出版《演讲》时也遇到了麻烦,他不久后告诉一位朋友:

It helped Hamilton’s case that he already had a reputation for mathematical genius, whereas Grassmann was an unknown country schoolteacher—it’s amazing that he managed to find a publisher at all for Ausdehnungslehre. Even Hamilton had trouble publishing his Lectures, as he told a friend shortly afterward:

它需要一定的科学声誉资本,这些声誉是在过去几年中积累起来的,这样才能使它成为危险的轻率行为。冒险出版一部虽然本质上相当保守,但却充满革命气息的作品。这是我不得不经历的磨难的一部分,是人生战斗中的一个插曲,我知道即使是坦率友好的人也会暗中,或者也可能是公开地谴责或嘲笑我,因为他们认为我的创新是可怕的。9

It required a certain capital of scientific reputation, amassed in former years, to make it other than dangerously imprudent to hazard the publication of a work which has, although at bottom quite conservative, a highly revolutionary air. It was a part of the ordeal through which I had to pass, an episode in the battle of life, to know that even candid and friendly people secretly, or, as it might happen, openly, censured or ridiculed me, for what appeared to them my monstrous innovations.9

他一定对这本书的反响感到失望。即使是约翰·赫歇尔,他在 1847 年听汉密尔顿讲授他的新发现时也非常兴奋,但他也写信给他说,《讲座》任何人读完都要花十二个月,消化则要花一生的时间。”六年后,当他再次阅读这本书时,他理解得稍微容易一些,但读了三章之后,他绝望地放弃了,因为他对书中分散注意力、独特的哲学感到失望。他建议汉密尔顿,如果他专注于更清楚地解释他的方法的数学算法和术语,那么读者可能会“更好地准备好接受你的形而上学解释。” 10

He must have been disappointed at the book’s reception. Even John Herschel, who had been so excited when he heard Hamilton lecture on his new discovery back in 1847, wrote to him saying the Lectures would “take any man a twelvemonth to read, and near a lifetime to digest.” He grasped it a little more easily when he came back to it six years later, but after three chapters he gave up in despair over the distracting, idiosyncratic philosophy. He advised Hamilton that if he focussed on explaining his method’s mathematical algorithms and terminology more clearly, then readers might be “better prepared to go along with you in your metaphysical explanations.”10

读者对格拉斯曼将哲学和数学奇怪地交织在一起感到更加意外,因为他自学数学的事实表明他不熟悉知名数学家使用的数学语言。在互联网出现之前,每当有新论文发表时,格拉斯曼就必须前往柏林寻找学术图书馆——如果他真的听说有论文需要跟进的话——即使在今天,从什切青到柏林最快的火车路线单程也需要近两个小时。这是另一个例子,说明为什么当时的数学进步往往出现在那些能够培养关系良好的学者群体的国家和机构中。

Readers were even less prepared for Grassmann’s strange interweaving of philosophy and mathematics, because the fact that he was self-taught in maths showed up in his lack of fluency in the mathematical language used by established mathematicians. In those days before the internet, every time a new paper was published Grassmann would have had to travel to Berlin to find an academic library—if indeed he ever heard there was a new paper to follow up—and even today, the fastest train route from Szcezcin to Berlin is almost a two-hour journey each way. It’s another example of why mathematical progress at that time tended to arise in those countries and institutions that were able to foster well-connected communities of scholars.

然而, 《解说》于 1844 年出版后,格拉斯曼立即将这本书的副本寄给了德高望重的高斯,他是当时德国数学界的领军人物,也许是当时世界上最优秀的数学家。高斯的回复相当冷淡,他说他已经研究了同样的想法半个世纪,并在 1831 年发表了其中一些。我提到了他的开创了复数的几何表示,格拉斯曼和汉密尔顿在发现时都不知道这一点,但正如我们将在第 10 章中看到的那样,高斯还开创了现在被称为“微分几何”的学科。格拉斯曼对“内”积和“外”积的研究也为这一分支做出了贡献,但在 1844 年,高斯觉得格拉斯曼的语言丛林中并没有什么新东西。在回复中,高斯说他很忙,需要时间来熟悉格拉斯曼的“特殊”术语——但显然,他一直没有时间去做。

Nevertheless, as soon as Ausdehnungslehre was published in 1844, Grassmann sent a copy to the venerable Gauss, the leading German mathematician and perhaps the best in the world at that time. Gauss replied rather dismissively, saying that he’d been working on the same ideas for half a century and had published some of them in 1831. I mentioned his pioneering geometrical representation of complex numbers, which neither Grassmann nor Hamilton had known of when they made their discoveries, but, as we’ll see in chapter 10, Gauss also initiated what is now known as “differential geometry.” Grassmann’s work on “inner” and “outer” products would also contribute to this branch, but in 1844 it seemed to Gauss that there was nothing new in Grassmann’s dense jungle of words. In his reply Gauss said that he was very busy, and that it would take time to familiarise himself with Grassmann’s “peculiar” terminology—but apparently, he never got around to it.

格拉斯曼没有气馁,他去了莱比锡,会见了高斯以前的学生莫比乌斯,他发表了一些早期的矢量思想。这再次让人想起了 19 世纪中叶的技术状况,因为当时没有电话或电子邮件,无法足不出户进行长距离通信。当然,旅行的好处是可以亲自接触,格拉斯曼的来访是数学“志趣相投的人”的一次欢乐聚会,正如莫比乌斯回忆的那样。格拉斯曼随后写了一封信,羞怯地问他的新朋友是否愿意评论《论数学》,因为他完全有能力评判这部作品的缺点和“书中可能包含的任何优点” 。11

Undeterred, Grassmann took a trip to Leipzig to meet Möbius, Gauss’s former student, who had published some early vectorial ideas. It’s another reminder of the state of technology in the mid-nineteenth century, for there were no telephones or email to make long-distance communications possible without having to leave home. The upside of having to travel, of course, is the personal contact, and Grassmann’s visit was a convivial meeting of mathematical “kindred spirits,” as Möbius recalled. Grassmann followed it up with a letter diffidently asking his new friend if he would review Ausdehnungslehre, since he would be well able to judge the work for both its weaknesses and “whatever merits the book may contain.”11

等待回复的过程一定令人焦急万分。事实上,可怜的格拉斯曼不得不等了四个月,因为莫比乌斯根本无法理解这样一本创新而又古怪的书。他在回复中说,他曾多次尝试阅读这本书,但每次都无法理解其中的哲学。然而,正如他告诉同事恩斯特·阿佩尔特,当他“强迫”自己浏览这本书时,他觉得格拉斯曼对几何加法和乘法的基本思想(我们今天所称的向量代数)的系统介绍可能对数学的发展有益。因此,正如他在回复中告诉格拉斯曼的那样,他请了一位熟悉哲学的数学家来审阅这本书(他称其为 Drobisch)。但为了以防万一,他建议格拉斯曼最好亲自审阅。12

It must have been a nail-biting wait for a reply. In fact, poor Grassmann had to wait four months, because Möbius simply couldn’t get his head around such an innovative and eccentric book. In his reply he said he’d made many attempts to read it, but each time he couldn’t get past the philosophy. Still, as he told his colleague Ernst Apelt, when he “forced” himself to skim through it, he felt that Grassmann’s systematic presentation of the fundamental ideas of geometric addition and multiplication—what we know today as the algebra of vectors—had something about it that might be good for the development of mathematics. So, as he told Grassmann in his reply, he’d asked a mathematician familiar with philosophy to review it (someone he referred to as Drobisch). But just in case, he suggested that it might be best for Grassmann to review it himself.12

莫比乌斯的哲学同事没有发表评论,格拉斯曼自己也没有勇气发表评论。相反,他将自己的书寄给了《数学与物理档案》的编辑约翰·格鲁纳特。和其他人一样,格鲁纳特也觉得这太难了,但他好心地建议格拉斯曼写一篇自己的评论摘要。于是,格拉斯曼就写了,尽管他的评论并不比这本书本身更有启发性。但至少他成功地将自己的作品介绍给了主流同行——他的评论是《Ausdehnungslehre》收到的唯一一篇评论。阿佩尔特告诉莫比乌斯,它太“奇怪”、太抽象、太不直观了——而阿佩尔特只读过格拉斯曼的评论!他当然不会开始读这本书。在莫比乌斯向海因里希·巴尔策推荐这本书后,他确实试着读了读,但每当他试图进入格拉斯曼的思维过程时,他都会“头晕目眩,眼前一片天蓝”。莫比乌斯回答说,他知道这种感觉!13

No review was forthcoming from Möbius’s philosophical colleague, and Grassmann was not so bold as to send out a review himself. Instead, he sent a copy of his book to Johann Grunert, the editor of Archiv der Mathematik und Physik. Like everyone else, Grunert found it too difficult, but he kindly suggested that Grassmann write his own review-summary. And so he did, although it was hardly more enlightening than the tome itself. But at least he’d managed to introduce his work to his mainstream colleagues—his was the only review Ausdehnungslehre received. It was simply too “strange,” too abstract and unintuitive, as Apelt told Möbius— and Apelt had only read Grassmann’s review! He certainly wasn’t about to start on the book. Heinrich Baltzer did try to read it after Möbius recommended it to him, but whenever he tried to enter into Grassmann’s thought-processes, he would “become dizzy and see sky-blue before my eyes.” Möbius replied saying he knew the feeling!13

事情就这样继续下去。但格拉斯曼是个斗士。和汉密尔顿一样,他坚信自己的方法将使物理学和几何学中的许多计算变得简单得多,并于 1845 年发表了一篇关于电动力学(即移动带电粒子和改变电磁场(我们现在称之为)的力学)的论文,该论文利用了他的新“外积”。电磁学是一个新颖而令人兴奋的话题,数学家们试图理解越来越多的实验结果——这是汉斯·奥斯特发现电磁学存在的二十五年后,但比詹姆斯·克拉克·麦克斯韦的电磁场理论早了二十年。麦克斯韦的理论将成为矢量历史的转折点,但与此同时,格拉斯曼意识到他的新系统可以更好地描述法国电磁学先驱安德烈-马里·安培于 1826 年发表的一项成果。该成果涉及小电路所施加的磁力的强度和方向——因此,你可以明白为什么矢量或格拉斯曼的“有向线”可能是它的理想语言。但是,当涉及到这种电磁现象的某些无法测试的方面时,格拉斯曼从已发表的实验结果中推导出的公式与安培给出的结果略有不同。细节在这里并不重要——首先,关于安培和格拉斯曼这两个公式中哪一个是更好的实验公式仍然存在争议契合;其次,格拉斯曼的矢量方法与麦克斯韦的方法更相符(关于麦克斯韦的方法我们稍后会详细介绍),因此,在 19 世纪 70 年代,随着麦克斯韦的胜利,格拉斯曼的版本终于获得认可,战胜了安培的版本。14

And so it went. But Grassmann was a battler. Like Hamilton, he was convinced that his method would make many computations in physics and geometry much simpler, and in 1845 he published a paper on electrodynamics—the mechanics of moving electric particles and changing electromagnetic fields (as we now call them)—which utilised his new “outer product.” Electromagnetism was a new and exciting topic, as mathematicians attempted to make sense of a growing number of experimental results—this was twenty-five years after Hans Øersted discovered the very existence of electromagnetism, but twenty years before James Clerk Maxwell’s theory of the electromagnetic field. Maxwell’s theory will prove a turning point in the story of vectors, but in the meantime, Grassmann realised that his new system could better describe a result published by the French pioneer of electromagnetism André-Marie Ampère, back in 1826. It concerned the strength and direction of the magnetic force exerted by a small electrical circuit—so you can see why vectors or Grassmann’s “directed lines” might be the ideal language for it. But the formula that Grassmann deduced from published experimental results gave some subtly different results from those given by Ampère when it came to certain untestable aspects of this electromagnetic phenomenon. The details do not matter here—except to say, first, that there is still contention over which of the two formulae, Ampère’s or Grassmann’s, is the better experimental fit; and, second, that Grassmann’s vectorial approach fitted better with Maxwell’s, of which more soon, and so Grassmann’s version won out over Ampère’s when it finally achieved recognition in the 1870s, in the wake of Maxwell’s triumph.14

幸运的是,尽管高斯的数学特质和创新成果不够完美,但这位才华横溢的外行人确实在早期获得了像莫比乌斯这样水平的支持者,莫比乌斯初出茅庐的矢量工作在射影几何和不变量研究中具有重要意义。莫比乌斯还以他非凡的“莫比乌斯带”预示了拓扑学领域的发展:一条两端相连的扭曲纸条,只有一侧,如果你沿着表面画一条线,就会看到这一点。高斯的另一位前学生约翰·林丁也在同一时间发现了这种现象。林丁创造了“拓扑学”一词,如今它指的是研究一般的不变性质,例如表面的边数或物体上的孔数:例如,甜甜圈和茶杯各有一个孔,如果它们都是用橡皮泥或面团制成的,你可以将一个变成另一个,而无需切割或粘合。换句话说,拓扑学并不关注甜甜圈或茶杯的具体形状,而是关注在这种连续的“橡皮泥”变形下不会改变的属性——它将在描述曲面和时空方面发挥作用。至于莫比乌斯,他显然在格拉斯曼的作品中看到了一些特别之处,即使他自己读不了多少,因为他在 1845 年初写信给格拉斯曼,告诉他有一场学术论文竞赛。这正是这位孤独的学校老师所需要的那种同事般的友善——论文主题正合他的胃口。

Luckily, despite his unpolished mathematical idiosyncrasies and innovative results, the gifted young outsider did find early supporters in people of the caliber of Möbius, whose fledgling vectorial work was important in projective geometry and the study of invariants. Möbius also prefigured the field of topology with his remarkable “Möbius band”: a twisted strip of paper joined at the ends that has only one side, as you can see if you trace a line along the surface. Another of Gauss’s former students, Johann Listing, also discovered this phenomenon, around the same time. Listing coined the term “topology,” which today refers to the study of general, invariant properties such as the number of sides of a surface, or the number of holes in an object: for instance, a donut and a teacup each have one hole, and if they were both made of plasticine or dough, you could turn one into the other without cutting or gluing. In other words, topology doesn’t focus on the specific shape of the donut or teacup but on properties that don’t change under this kind of continuous “plasticine” deformation—and it will prove useful in describing curved surfaces and space-times. As for Möbius, he evidently saw something special in Grassmann’s work even if he himself couldn’t read much of it, for he wrote to Grassmann in early 1845 to tell him about an academic essay competition. It was just the sort of collegial kindness the isolated schoolteacher needed—and the essay topic was right up his alley.

在第 3 章中,我提到了牛顿在用大小和方向表达物理量方面的开创性作用,但牛顿的宿敌莱布尼茨也在其中扮演了重要角色。这与符号的力量有关。莱布尼茨说:“值得注意的是,符号有助于发现。这以一种最奇妙的方式减少了思维劳动。”我们在微积分中看到了这一点,莱布尼茨的微分学dy / dx符号比牛顿的更适合计算和概念理解——约翰·沃利斯在托马斯·哈里奥特的代数本身可以充分利用符号。但莱布尼茨也希望为几何学提供一种类似的省力语言。15

In chapter 3, I mentioned Newton’s pioneering role in expressing physical quantities in terms of both magnitude and direction, but Newton’s nemesis Leibniz also has a part to play in the story. It has to do with the power of symbols. “It is worth noting,” Leibniz said, “that notation facilitates discovery. This, in a most wonderful way, reduces the mind’s labors.” We’ve seen this in calculus, where the Leibnizian dy/dx notation for differential calculus is better adapted to computations and conceptual understanding than Newton’s —and John Wallis had seen it in Thomas Harriot’s full use of symbols in algebra itself. But Leibniz wanted a similar labour saving language for geometry, too.15

他在 1679 年写给荷兰物理学家克里斯蒂安·惠更斯的一封信中表达了这一想法。惠更斯是光的波动理论和许多其他事物(包括摆钟)的先驱。(虽然摆钟在今天看来似乎是一件古雅的遗物,但它是精确计时的一大进步。半个世纪前,在他们进行下落运动实验时,伽利略不得不用滴水的“水钟”来计时,而哈里奥特则用他的脉搏来计时。)莱布尼茨向惠更斯建议,需要一种新的代数形式,一种“明显是几何或线性的代数,能够直接表达情形,就像(普通)代数直接表达量级一样。”(他所说的“情形”是指空间中的相对位置。)笛卡尔率先使用坐标来描述此类位置,但莱布尼茨在寻找某种更清晰的几何形式——事后看来,我们可以将其想象成一种独立于坐标的矢量箭头或格拉斯曼有向线的语言。莱布尼茨本人并没有找到这样的系统,他写给惠更斯的信直到 1833 年才发表——此时莫比乌斯和贝拉维蒂斯已经在矢量方向上迈出了试探性的一步。但 1845 年的征文竞赛旨在挑战数学家更全面地回答莱布尼茨的建议,奖项将于 1846 年颁发,以纪念莱布尼茨 200 岁诞辰。事实证明,挑战是如此巨大,以至于格拉斯曼是唯一一个参赛的人。这是他发表《引言》之后的第一个好运,因为他的获奖论文发表于 1847 年,附录由莫比乌斯撰写,莫比乌斯的名气和人脉关系使其他数学家更有可能注意到他的门生。

He expressed the idea in a 1679 letter to the Dutch physicist Christian Huygens, pioneer of the wave theory of light and many other things— including the pendulum clock. (Though it seems a quaint relic today, the pendulum clock was a great advance in accurate timekeeping. In their experiments on falling motion half a century earlier, Galileo had had to measure time by a dripping “water clock” and Harriot had used his pulse.) Leibniz suggested to Huygens that a new form of algebra was needed, one that is “distinctly geometrical or linear and which will express situation directly as [ordinary] algebra expresses magnitude directly.” (By “situation” he meant relative position in space.) Descartes had spearheaded the use of coordinates to describe such positions, but Leibniz was looking for something more clearly geometrical—something that we, with hindsight, might imagine as a language of vectorial arrows or Grassmannian directed lines, independent of coordinates. Leibniz himself didn’t find such a system, and his letter to Huygens was published only in 1833—by which time Möbius and Bellavitis had already taken tentative steps in the vectorial direction. But the 1845 essay competition was designed to give mathematicians the challenge of answering Leibniz’s suggestion more fully, and the prize was to be awarded in 1846, in honour of Leibniz’s two hundredth birthday. As it turned out, the challenge was so great that Grassmann was the only one who entered. It was his first piece of luck after publishing Ausdehnungslehre, for his prize-winning paper was published in 1847, with an appendix by Möbius, whose fame and connections made it more likely that other mathematicians would take notice of his protégé.

这篇文章的可读性并不比《论证》高,尽管它确实有助于将格拉斯曼的书带入更广泛的数学界。但当汉密尔顿终于在 1850 年左右听说这本书时,它以难度高而闻名,他觉得自己可能得学会抽烟才能读完它!然而,当他在 1852 年秋天开始阅读它时,他告诉德摩根这是“一部非常原创的作品”,“如果德国人认为我值得关注,他们可能会建立在我看来,格拉斯曼的思想与我的思想相竞争,但直到我的观点形成并发表很久之后,我才意识到这一点。”三个月后,他告诉德·摩根,他仍然在“怀着极大的钦佩和兴趣”阅读格拉斯曼的作品,尽管有时他发现用德语阅读很困难。他很自然地感到惊讶,格拉斯曼“以最明显和最完美的独立性”找到了与他的四元数如此相似的系统。16

Not that the essay was much more readable than Ausdehnungslehre, although it did help to bring Grassmann’s book into the wider mathematical community. But when Hamilton finally heard of it, sometime around 1850, such was its reputation for difficulty that he thought he might have to learn to smoke in order to get through it! When he began to read it in the fall of 1852, however, he told De Morgan it was “a very original work,” which “the Germans, if they think me worth noticing, will perhaps set up in rivalship with mine, but which I did not see till long after my own views were formed and published.” Three months later, he told De Morgan he was still reading Grassmann “with great admiration and interest,” although some days he found it hard to read in German. He was amazed, naturally enough, that Grassmann had, “with the most obvious and perfect independence,” found such a similar system to his quaternions.16

两天后,他的热情有所减退,但他再次写信给德·摩根,说他认为格拉斯曼是“一位伟大的、最具德国特色的天才”。但他很难理解格拉斯曼所有不同类型的乘法(他有比我们在向量张量代数中学到的两种更多的乘法)。“我认为我确实理解了他的积,这对于一个没有学会抽烟的人来说已经说明了些什么。甚至他的内积……我也能很好地接受[因为它们]与我的四元数的‘标量部分’非常相似,他的‘外积’与我的‘矢量部分’非常相似。”(正如我所提到的,汉密尔顿的四元数乘积的“标量和矢量部分”是我们矢量的“标量和矢量积”。)17

Two days later, his enthusiasm had dimmed somewhat, but he wrote to De Morgan again, saying that he considered Grassmann “a great and most German genius.” But he was having trouble grasping all of Grassmann’s different kinds of multiplication (he had many more than the two that have come down to us in vector-tensor algebra). “His outer products I think I do understand, and that is saying something for a person who has not learned to smoke. And even his inner products … I can swallow pretty well [for they] have much analogy to my ‘scalar parts’ of a quaternion, and his ‘outer products’ to my ‘vector parts.’ ” (As I mentioned, Hamilton’s “scalar and vector parts” of a product of quaternions are our “scalar and vector products” of vectors.)17

德·摩根对汉密尔顿在代数理论方面的进展非常着迷,一周后他又收到了更新。汉密尔顿说,他当时正和女儿一起读书,女儿从他身后看着他,“对哲学家的愚蠢感到好笑”,这时他注意到,虽然他自己是从约翰·沃伦那里得到了添加“线条”的想法,但格拉斯曼似乎是自己想出来的。(他不知道贾斯图斯·格拉斯曼——直到今天也没有人知道贾斯图斯是如何想到几何代数的。)然而,汉密尔顿继续说,格拉斯曼花了 139 页“鸵鸟般的胃”才切中要点——有向线(或矢量)的实际代数。汉密尔顿说,他很早就想到了这一点,并把它作为他在 1848 年关于新系统的公开演讲的起点。大概他认为这样做可以让听众不需要铁打的“鸵鸟”胃来消化这种新数学。尽管如此,这些讲座的出版版本仍然很难读懂,因为汉密尔顿非常小心地解释他的新代数学中的每一个基本步骤,以至于他使用了大量奇怪的术语——包括他所用的所有新词,比如格拉斯曼 (Grassmann) 曾创造过这种代数。但他确实有一种对话式的风格,他还用实际的例子来说明他的新向量代数——特别是来自天文学和他作为爱尔兰皇家天文学家的工作。18

De Morgan was fascinated by Hamilton’s progress with Ausdehnungslehre, and a week later he received another update. Hamilton said that he’d been reading with his daughter looking over his shoulder, “amused at the folly of philosophers,” when he noticed that while he himself had got the idea of adding “lines” from John Warren, Grassmann seemed to have worked it out himself. (He didn’t know about Justus Grassmann—and even today no one knows how Justus came to his idea of geometrical algebra.) However, Hamilton continued, Grassmann took 139 “ostrich-stomachneeding” pages to get to the point—the actual algebra of directed lines (or vectors). Hamilton had conceived this early, he said, and had taken it as the starting point of his 1848 public lectures on his new system. Presumably he thought he was thereby sparing his listeners the need for a cast-iron “ostrich” stomach to digest this new mathematics. Still, the published version of those lectures was tough going, because Hamilton took so much care to explain each fundamental step in his new algebra that he employed an excess of strange terminology—including all the new words that he, like Grassmann, had coined. But he did have a conversational style, and he also illustrated his new vector algebra with practical examples—particularly from astronomy and his work as Ireland’s royal astronomer.18

在听说格拉斯曼之前,汉密尔顿就开始将这些讲座扩展成一本书,由此产生的《四元数讲座》在他给德·摩根的一系列信件后不久出版。汉密尔顿使用希腊字母来区分矢量和数字,数字通常用拉丁字母表示——在他的序言第 62 页,在区分“普通代数”(其中ab = ba)和他的新非交换矢量代数(其中 αβ = −βα)时,汉密尔顿承认格拉斯曼提出了他自己的“一种非常新颖和非凡的斜线(或有向线)非交换乘法”。但他明确表示,他自己的工作与“深刻的哲学家”格拉斯曼的工作无关,他指出格拉斯曼无法将二维复平面扩展到整个空间。

Hamilton had begun expanding these lectures into a book before he’d heard of Grassmann, and the resulting Lectures on Quaternions was published not long after his series of letters to De Morgan. Hamilton used Greek letters to distinguish vectors from numbers, which are usually signified by Latin letters—and on page 62 of his preface, when distinguishing “ordinary algebra,” where ab = ba, from his new noncommutative vector algebra, where αβ = −βα, Hamilton acknowledged that Grassmann had come up with his own “species of non-commutative multiplication for inclined [or directed] lines in a very original and remarkable work.” But he made it clear that his own work was independent of the “profound philosopher” Grassmann’s, and he noted that Grassmann had not been able to extend the two-dimensional complex plane to all of space.

这是两次发现之间的区别。汉密尔顿的四元数主要用于处理三维几何运算,如旋转,正如我所提到的,它们已经找到了一些非凡的应用。格拉斯曼的系统更加抽象,这使得它在三维物理的直接应用中不如矢量和四元数有用,但在后来的推广中更有用,最终导致了现代张量分析。

This is the nub of the difference between the two discoveries. Hamilton’s quaternions were designed largely to handle algebraically threedimensional geometrical operations such as rotations, and they have found some extraordinary applications as I’ve mentioned. Grassmann’s system was more abstract, which made it less useful than vectors and quaternions in immediate applications in 3-D physics, but more useful in the later generalisations that would ultimately lead to modern tensor analysis.

当然,汉密尔顿或格拉斯曼当时并没有预见到这一切。其他人也没有。尽管汉密尔顿早年就声名鹊起,但在 19 世纪 50 年代和 60 年代初,甚至连他最初将复数表示为实数的有序对的想法都没有实现,更不用说四元数了。格拉斯曼的想法甚至没有取得任何进展。但他和汉密尔顿从未放弃。我之前提到过汉密尔顿 1859 年写给新同事 (Peter Guthrie Tait) 的一封欣喜若狂的信,信中谈到他们在将四元数应用于物理学方面的进展,“还有什么比这更简单或更令人满意的吗?你不觉得,也不认为,我们走在正确的道路上,以后会得到感谢吗?别在意什么时候……”格拉斯曼也有类似的感受:在 1862 年版的《解说》中——以更数学、更严谨的方式重新编写,更少的哲学风格,加上一些新材料——他在序言的结尾写道:“我完全相信,我在这里介绍的科学上所付出的劳动不会白费,这些劳动占据了我生命的大部分时间,并耗费了我最艰苦的努力。”他希望不要显得傲慢,但他也希望他的想法最终能够结出硕果,无论它们被搁置多久。

Not that Hamilton—or Grassmann—foresaw all this at the time, of course. Neither did anyone else. Despite Hamilton’s earlier fame, in the 1850s and early 1860s not even his initial idea of representing complex numbers as ordered pairs of real numbers had taken off, let alone his quaternions. Grassmann’s ideas had made even less headway. Yet he and Hamilton never gave up. I mentioned earlier Hamilton’s joyous 1859 letter to a new colleague (Peter Guthrie Tait), saying of their progress in applying quaternions to physics, “Could anything be simpler or more satisfactory? Do you not feel, as well as think, that we are on a right track, and shall be thanked hereafter? Never mind when....” Grassmann felt similarly: in the 1862 version of Ausdehnungslehre—a reworking in a more mathematical, less philosophical style, plus some new material—he ended the foreword by saying, “I remain completely confident that the labor which I have expended on the science presented here and which has demanded a significant part of my life as well as the most strenuous application of my powers, will not be lost.” He hoped not to seem arrogant, but he also hoped his ideas would eventually bear fruit, no matter how long they might lie dormant.

他们确实沉寂了下来,因为第二版《Ausdehnungslehre 》的成绩并不比第一版好。

And lie dormant they did, for the second version of Ausdehnungslehre fared little better than the first.

创意发酵

CREATIVE FERMENT

数学创造力与其他形式并无不同,在艺术、音乐和文学领域,新的形式和风格也往往需要一段时间才能流行起来。但行之有效的东西会逐渐被接受,尤其是当一系列新艺术家和艺术形式开始融合或交叉融合时。事实上,当汉密尔顿和格拉斯曼一直在研究一种使用代数运算和符号处理几何的新方法时,几何领域也发生了另一场数学革命。

Mathematical creativity is not unlike any other form, and in art, music, and literature, new forms and styles often take a while to catch on, too. But things that work gradually become accepted, especially when a range of new artists and art forms begins to meld or cross-pollinate. Indeed, while Hamilton and Grassmann had been working out a new way of handling geometry using algebraic operations and symbolism, another mathematical revolution had been taking place in geometry.

欧几里得几何学非常严谨,在两千多年中一直占据主导地位。问题是,它是一门日常“平面”空间的科学:例如书页、石板、桌面、沙地——任何欧几里得都能在其上画出永不相交的平行线和内角和为 180° 的三角形的表面。但很明显,在球体或地球仪等曲面上,两条平行线确实相交。看看两条相邻的经线穿过赤道时的样子:它们一开始是平行的,但随着向北移动,它们逐渐靠近,直到在北极相交。因此,当汉密尔顿和格拉斯曼发现可以放弃交换律而采用乘法,同时仍然拥有一致的代数时,其他数学家正在通过放弃欧几里得所谓的平行公设来开发新的“非欧几里得”几何。

Euclidean geometry was beautifully rigorous and had reigned supreme for more than two thousand years. Trouble was, it was a science of everyday “flat” space: the pages of a book, for example, or a slate, a tabletop, a patch of sand—any surface on which Euclid could draw such things as parallel lines that never meet and triangles whose angles add up to 180°. But it was clear that on a curved surface such as a ball or the globe of Earth, two parallel lines do meet. Check out two nearby lines of longitude as they cross the equator: they start off parallel, but as you move north, they gradually move closer together until they meet at the North Pole. So, while Hamilton and Grassmann were discovering that you could ditch the commutative law for multiplication and still have a consistent algebra, other mathematicians were developing new “non-Euclidean” geometries by abandoning Euclid’s so-called parallel postulate.

其中最主要的是德国的高斯、匈牙利的亚诺什·波约伊和俄罗斯的尼古拉·罗巴切夫斯基,他们的发现将这对张量的故事产生了影响——这个故事包括爱因斯坦现在著名的弯曲时空。与此同时,随着非欧几里得几何在 19 世纪 60 年代渗入主流,人们对非交换代数的兴趣也越来越浓厚——也对汉密尔顿将复数作为实数的有序对的代数处理产生了兴趣,而不再是它们在阿尔冈平面上的“欧几里得”几何表示。

Chief among them were Gauss in Germany, Janos Bolyai in Hungary, and Nicolai Lobachevsky in Russia, and their discoveries will have ramifications for the story of tensors—a story that includes Einstein’s nowfamous curved space-time. Meantime, as non-Euclidean geometry filtered into the mainstream in the 1860s, so people became more interested in noncommutative algebras—and also in Hamilton’s algebraic treatment of complex numbers as ordered pairs of real numbers, moving on from their “Euclidean” geometric representation on the Argand plane.

数学和数学家们又花了 60 年的时间才变得足够成熟,不仅发明了张量分析,而且重新审视了“解释学”,并在其抽象性中认识到了新的张量工具。但格拉斯曼等了这么久,在 19 世纪 60 年代和 70 年代,他基本上放弃了数学;他写的几篇论文没有达到他早期的标准,好像他的心已经离开了数学。相反,他把自己巨大的精神能量投入到编写与他的教学相关的科目(德语、拉丁语、数学、音乐、宗教和植物学)的教科书和论文中。和汉密尔顿一样,他也学习了梵语。19 世纪 70 年代,他翻译了 1123 页的《梨俱吠陀》,并出版了自己关于这部印度教经典的更长的译本——这项艰巨的劳动使他当之无愧地获得了美国东方学会会员和 1876 年图宾根大学的荣誉博士学位。他完成了所有这些工作,包括教科书和语言文学,以及大部分数学和物理,同时还从事全职教学并抚养 11 个孩子(其中 4 个已安葬)——他于 1849 年结婚,那时他首次出版了《离题说》五年后。

It would take yet another sixty years for mathematics, and mathematicians, to become sophisticated enough not only to invent tensor analysis, but also to revisit Ausdehnungslehre and to recognise in its very abstractness new tensorial tools. But that was far too long for Grassmann to wait, and in the 1860s and 1870s he turned mostly away from mathematics; the few papers he wrote were not up to his earlier standard, as if his heart had gone out of it. Instead, he put his prodigious mental energy into writing textbooks and papers on subjects related to his teaching—German, Latin, maths, music, religion, and botany. Like Hamilton, he’d also learned Sanskrit, and in the 1870s he made a 1,123-page translation of the Rig Veda and published his own even longer work on the Hindu classic—exhausting labour that deservedly earned him membership in the American Oriental Society and an honorary doctorate from the University of Tübingen, in 1876. He did all this, the textbooks and the philology, as well as much of his maths and physics, while teaching full time and raising eleven children (and burying four of them)—he’d married in 1849, five years after he first published Ausdehnungslehre.

为了便于理解政治背景,我还要补充一点,格拉斯曼除了在 1848-49 年求爱和结婚外,还和兄弟一起出版了一份每周的政治报纸——格拉斯曼希望看到德国在君主立宪制下统一,但当时,奥托·冯·俾斯麦等人反对宪法改革。经过几次领土战争和二十年,新的德意志帝国才得以统一——俾斯麦担任总理。格拉斯曼创办报纸时,爱尔兰最紧迫的政治问题是日益高涨的罗马天主教独立运动——但汉密尔顿是坚定的圣公会教徒(当时爱尔兰的国教,因为他是一位爱尔兰学者(1891年),爱尔兰当时还是英国的一部分。他没有把精力投入政治,而是致力于推动科学,担任爱尔兰皇家科学院院长,并为英国科学促进会做出积极贡献。

For some political context, I should add that as well as courting and marrying in 1848–49, Grassmann had joined his brother in publishing a weekly political newspaper in those years—Grassmann wanted to see Germany unified under a constitutional monarchy, but at the time, the likes of Otto von Bismarck opposed constitutional reform. It would take several territorial wars and two decades before unification of a new German empire was achieved—with Bismarck as chancellor. While Grassmann was producing his newspaper, the most pressing political issue in Ireland was the growing movement for Roman Catholic independence—but Hamilton was a staunch Anglican (the national religion of Ireland at that stage, for the country was still part of the United Kingdom). He put his energy not into politics but into the promotion of science, through his role as president of the Royal Irish Academy, and his active contributions to the British Association for the Advancement of Science.

• • •

• • •

汉密尔顿从未放弃对四元数应用的追求。除了《四元数讲座》之外,他还发表了 100 多篇有关该主题的论文。尽管他声名显赫,数学实力雄厚,但仍有许多人认为,他几乎将生命的最后 20 年全部奉献给了四元数,这是自欺欺人。人们普遍认为,四元数虽然很出色,但用处不大。

Hamilton never gave up his pursuit of quaternion applications. Aside from his Lectures on Quaternions, he published more than a hundred papers on the subject. Despite his fame and mainstream mathematical prowess, there were many who thought he was deluding himself by devoting the last twenty years of his life almost solely to quaternions. There was a widespread feeling that quaternions were brilliant, but they weren’t all that useful.

然而,随着新一代的崛起,以及麦克斯韦和彼得·格思里·泰特进入我们故事的下一章,汉密尔顿的命运开始改变。泰特成为四元数的坚定支持者,其他人正是通过他和麦克斯韦的工作发展了今天大学里教授的现代矢量分析。可怜的格拉斯曼的思想将继续被证明过于笼统——而且很难与他的“奇怪”哲学和“特殊”术语区分开来——无法直接产生影响。我们稍后会再次与他联系,但即便如此,矢量分析和非欧几里得几何最初将我们引向张量。19

Hamilton’s fortunes began to change, however, with the rise of a new generation, and the entrance of Maxwell and Peter Guthrie Tait in the next chapter of our story. Tait became a fierce supporter of quaternions, and it was largely through his and Maxwell’s work that others would develop the modern vector analysis taught in university today. Poor Grassmann’s ideas would continue to prove too general—and too difficult to disentangle from his “strange” philosophy and his “peculiar” terminology—to be a direct influence. We’ll check in with him again later, but even so, it is vector analysis and non-Euclidean geometry that will initially lead us to tensors.19

(6)泰特和麦克斯韦

(6) TAIT AND MAXWELL

绘制电磁矢量场

Hatching the Electromagnetic Vector Field

尽管赫尔曼·格拉斯曼的作品难以理解且过于抽象,但威廉·罗文·汉密尔顿而非格拉斯曼成为当今教授的矢量分析的直接联系者,这在一定程度上也是历史的偶然。因为彼得·格思里·泰特和詹姆斯·克拉克·麦克斯韦在学生时代相识,无疑是命运的安排。

Although Hermann Grassmann’s work was difficult to penetrate and excessively abstract, it is also partly an accident of history that William Rowan Hamilton, not Grassmann, is the direct link to the vector analysis taught today. For surely it was fate that brought Peter Guthrie Tait and James Clerk Maxwell together as schoolboys.

他们均于 1831 年出生于苏格兰——比汉密尔顿宣布他在四元数问题上取得的突破早 12 年——从十岁起,他们都进入了著名的爱丁堡学院学习。对于没有母亲的马克斯韦尔来说,最初的经历并不愉快。他是一个古怪的孩子,由溺爱他、不循规蹈矩的父亲抚养长大。他的父亲是一名律师,也是格伦莱尔的领主,格伦莱尔是爱丁堡西南约 90 英里的一处家族庄园。因此,当他来到这所大城市的学校时,马克斯韦尔是个怪人,幽默感古怪,说话犹豫不决,穿着舒适而非时尚的自制衣服——因为所有这些违反班级规范的行为,他受到了无情的欺负。他的学习成绩平平无奇:作为一个自主学习者,他最初觉得学校很无聊;再加上他的语速缓慢、性格腼腆,他得到了“傻瓜”的绰号,所以你可以想象,这个绰号是如何让他成为一个“等着”被打的男孩的。1

They were both born in Scotland in 1831—twelve years before Hamilton announced his quaternion breakthrough—and from the age of ten they both attended the prestigious Edinburgh Academy. Not that it was a happy experience for the motherless young Maxwell at first. He was an eccentric child, being raised by his doting and unconventional father, a barrister and the Laird of Glenlair, the family estate some ninety miles southwest of Edinburgh. So when he arrived at the big city school, Maxwell was an odd figure, with a quirky sense of humour, a hesitant way of speaking, and homemade clothes that were comfortable rather than fashionable— and for all these transgressions of the class code he was mercilessly bullied. It didn’t help that his academic performance was indifferent: as a selfdirected learner he initially found school boring; this, together with his slow speech and shyness, led to the nickname “Dafty,” so you can imagine how that singled him out as a boy “waiting” to be punched.1

对他和科学界的未来来说幸运的是,他温和的同学刘易斯·坎贝尔与麦克斯韦成为了朋友,最终他得以安定下来,与坎贝尔和泰特一起成为尖子生。很明显,这是一场富有成效的友好竞争,泰特和麦克斯韦很快也成为了好朋友。和麦克斯韦一样,泰特也失去了父母——他的父亲曾是一位公爵的秘书——并从乡下搬到了爱丁堡,尽管他的家乡达尔基斯离这里只有八英里。

Fortunately for him, and for the future of science, his gentle classmate Lewis Campbell befriended Maxwell, so that eventually he was able to settle down and become one of the top students, along with Campbell and Tait. Tellingly, it was a productive and friendly rivalry, and Tait and Maxwell soon became firm friends, too. Like Maxwell, Tait had lost a parent—his father, who’d been secretary to a duke—and had moved to Edinburgh from the country, although his hometown of Dalkeith was only eight miles away.

爱丁堡学院是一所相对进步的学校,泰特、麦克斯韦和坎贝尔在那里表现出色。16 岁时,他们都考上了爱丁堡大学,但一年后坎贝尔转学到牛津大学——他成为了一名古典学家和英国国教牧师,后来成为他童年好友麦克斯韦的一位极具同理心的传记作者——而泰特则转学到剑桥大学,在那里他以 1852 年毕业的高级考官的身份超越了所有同龄人。高级考官是 Tripos 考试中成绩最好的学生,这是一项为期八天的严酷数学考试,包括大约十六场难度递增的考试。数学被视为所有学生的重要心理准备,无论他们的最终职业是什么——如果你能成为得分最高的考官之一,你就一定会有一条良好的职业道路。泰特非常激动,他先是把这个消息电报给家人,然后给他以前的爱丁堡老师写信,“我激动得说不出话来,就像歌里唱的:我是高级牧马人!”他还获得了久负盛名的史密斯奖,该奖是针对更为复杂的数学考试中成绩最优秀的学生颁发的——总的来说,这是一项非常了不起的成就,他和麦克斯韦的母校为他举行了特别的庆祝活动。2

The Edinburgh Academy was a relatively progressive school, and Tait, Maxwell, and Campbell excelled there. Then, at sixteen, they all went on to Edinburgh University, although after a year Campbell transferred to Oxford—he became a classicist and an Anglican minister, and later a wonderfully empathetic biographer of his childhood friend Maxwell—while Tait transferred to Cambridge, where he outshone all his peers, graduating as the Senior Wrangler of 1852. The Senior Wrangler was the student with the best marks in the Tripos, a grueling series of around sixteen increasingly difficult mathematics examinations, held over eight days. Mathematics was seen as important mental preparation for all students no matter their ultimate profession—and if you managed to be one of the highest scoring wranglers you were assured of a good career pathway. Tait was so excited that first he telegraphed the news to his family, and then he wrote to his erstwhile Edinburgh teachers, “I’m all in a flutter, I scarcely can utter, etc, as the song has it: i am senior wrangler!” He also took out the prestigious Smith’s Prize, for the top student in an even more complex series of maths exams—and all up it was such a fine achievement that his and Maxwell’s old school held a special celebration in his honour.2

毕业后,泰特获得了剑桥大学的奖学金——这是成为顶尖论文写作者的好处之一,因为这让他获得了一份工作和教学经验,这很可能让他以后在剑桥大学或其他一流大学获得永久讲师职位。不过,目前,他沉浸在新获得的安全感中,没有了备考的苦差事。他甚至给自己买了一本刚刚出版的汉密尔顿的《四元数讲座》 ——他不知道这本书讲的是什么,但他对“四元数”这个奇怪的名字很感兴趣。他在一次旅行中读了前六章暑假期间,麦克斯韦参加了一次射击之旅,但回到剑桥后,他的教学和研究占据了他所有的时间,他把四元数放在了一边。(相比之下,麦克斯韦拒绝打猎或钓鱼,因为他热爱动物。他特别擅长与马打交道,身边总是有一只心爱的狗。)

After graduating, Tait won a fellowship at Cambridge—one of the benefits of being a top wrangler, for it gave him a job and teaching experience that would presumably lead to a permanent lectureship later, at Cambridge or some other leading institution. For the moment, though, he reveled in his newfound security, free of the grind of exam preparation. He even treated himself to a copy of Hamilton’s Lectures on Quaternions, hot off the press—he had no idea what it was about, but he was intrigued by the strange name “quaternion.” He read the first six chapters while away on a summer holiday shooting trip, but once back at Cambridge, his teaching and research took up all his time, and he laid Quaternions aside. (Maxwell, by contrast, refused to hunt or fish, such was his love of animals. He was especially brilliant with horses, and he always had a much-loved dog by his side.)

泰特在剑桥大学就读期间,麦克斯韦留在了爱丁堡大学——显然是为了取悦他的父亲——但那里优秀而广泛的教学为他提供了完美的训练场,让他能够运用实践和哲学技能,很快他便凭借这些技能改变了物理学。1850 年,在爱丁堡大学获得学位后,他进入剑桥大学继续深造。虽然花了一段时间才适应,但腼腆、笨拙的年轻人达夫蒂很快结识了“一群”新朋友,因为剑桥大学的人都觉得他“和蔼可亲、风趣幽默”,能够以诙谐、博学的方式谈论任何话题。一位新认识的人在交谈后写道:“我从未见过像他这样的人。”另一位后来回忆道:“在三一学院(麦克斯韦在剑桥大学的学院)认识他的每个人都能回忆起他的一些善举或一些行为,这些都给他留下了不可磨灭的善良印象。” 3

While Tait was at Cambridge, Maxwell had stayed on at Edinburgh— apparently to please his father—but the excellent, broad-ranging teaching there proved the perfect training-ground for the practical and philosophical skill with which he would soon transform physics. Then, after finishing his Edinburgh degree in 1850, he enrolled at Cambridge to further his scientific studies. It took a while to settle in, but soon the shy, awkward young Dafty made “a troop” of new friends, for the Cambridge crowd found him “genial and amusing,” able to converse in a witty and erudite way on any subject at all. “I never met a man like him,” one new acquaintance recorded after such a conversation, while another later reminisced, “Everyone who knew him at Trinity [Maxwell’s college at Cambridge] can recall some kindness or some act of his which has left an ineffaceable impression of his goodness.”3

虽然麦克斯韦对自己的新社交生活感到高兴,但他对当时的剑桥课程并不满意,因为该课程强调技术熟练度而不是深度思考。年底,当他准备考试时,他在一首长诗《牧马人的幻象》中表达了自己的沮丧。以下是他的幽默和他对考试制度的看法:

While Maxwell delighted in his new social life, he was not so impressed with the Cambridge curriculum of the time, with its emphasis on technical proficiency rather than deep thinking. At the end of the year as he was preparing for exams, he expressed his frustration in a long poem, “A Vision of a Wrangler.” Here’s a taste of his humour, and his view of the exam system:

壁炉里闪烁的余烬

这表明十一月是多么的沉闷

雾气笼罩了我麻木的肢体

就像一只拔了毛又瘦的鹅。

当我准备睡觉的时候,我

我颤抖着声音问自己,

如果我读了所有的东西,我

曾经有过丝毫使用。4

In the grate the flickering embers

Served to show how dull November’s

Fogs had stamped my torpid members

Like a plucked and skinny goose.

And as I prepared for bed, I

Asked myself with voice unsteady,

If of all the stuff I read, I

Ever made the slightest use.4

这只是 24 首越来越博学的诗中的第二首,这些诗围绕着他邪恶的新母校的奇幻化身展开,他表达了对一个要求死记硬背的学习制度的愤慨,这个制度要求学生死记硬背,然后给最优秀的学生提供舒适的学术工作——这些争论者随后成为维持这个制度的精英阶层的一部分。这首诗以一个严肃的精神宣言结束,即最好是默默地思考“创造的荣耀”,而不是漫无目的地研究物理学家用来表示自然的超然的数学符号。他写这首非凡的诗时才 21 岁;然而,在 12 年后,他将展示自然的符号表示有多么奇妙。后来,他自己也成为了一名考官和主持人,他将在使 Tripos 问题(以及剑桥教学大纲)与正在发现的令人兴奋的新科学更加相关方面发挥主导作用。5

This is but the second of twenty-four increasingly erudite verses centred on a fantastical incarnation of his evil new alma mater, as he expresses his outrage at a system that demands slavish rote study and then rewards its best students with cushy academic jobs—these wranglers then becoming part of the elite who keep the system going. The poem ends with a serious spiritual declaration, that it is better to silently contemplate the “glories of Creation” than to mindlessly study the detached mathematical symbols that physicists use to represent nature. He was twenty-one when he wrote this extraordinary poem; in twelve years’ time, however, he would show just how wondrous the symbolic representation of nature can be. Later, he would also become an examiner and moderator himself, and he would play a leading role in making the Tripos questions—and therefore the Cambridge syllabus—much more relevant to the exciting new science being discovered.5

然而,1853 年 11 月,随着期末考试的临近,他又写了一长诗——《在十一月激情熄灭后读数学是不明智的信念下写下的诗句!》在诗中,他谈到了年轻人承受的沉重考试压力,以及将学术奖项和学位误认为智慧的愚蠢行为。因此,他更喜欢学习自己感兴趣的东西,而不是死记硬背——泰特对麦克斯韦在大学荣誉学位考试中准备不足感到惊讶。尽管如此——多亏了麦克斯韦“纯粹的智力”,泰特说道——他还是以第二名的成绩毕业,与高级名次 (EJ Routh) 并列史密斯奖。他的父亲在一封感人至深的支持信中表达了自己的骄傲:“[你的表弟乔治] 凌晨 2 点来到我的房间……看了特快列车送来的《星期六泰晤士报》 ,我在早餐前收到了你的来信……虽然你在冠军 [史密斯] 的比赛中与高级牧马人不相上下,但你只比他落后一点点 ”至于麦克斯韦的前学校数学老师,他“欣喜若狂”,就像他为泰特的成功而兴奋一样。6

In November 1853, however, as his final exams drew near, he wrote another long poem—“Lines Written under the Conviction That It Is Not Wise to Read Mathematics in November after One’s Fire Is Out!”—in which he speaks of the soul-crushing exam pressure on young people, and the folly of mistaking academic prizes and degrees for wisdom. So mostly he preferred to study what interested him instead of cramming—Tait was amazed at how unprepared Maxwell was for the Tripos. Still—thanks to Maxwell’s “sheer strength of intellect,” as Tait put it—he managed to graduate as Second Wrangler, tying with the Senior Wrangler (E. J. Routh) for the Smith’s Prize. His father expressed his pride in a touchingly supportive letter: “[Your cousin George] came into my room at 2 am … having seen the Saturday Times, received by express train, and I got your letter before breakfast.... As you are equal to the Senior Wrangler in the champion [Smith’s] trial, you are but a very little behind him.” As for Maxwell’s former school maths teacher, he was “beside himself ” with excitement, just as he’d been with Tait’s success.6

你可以从 1854 年 2 月麦克斯韦参加的试卷的第 8 个问题中感受到史密斯奖考试的难度:它要求证明现在被称为“斯托克斯定理”的定理——爱尔兰出生的过去几年,数学家乔治·斯托克斯一直在编写史密斯奖论文。1854 年的论文是该定理首次以印刷形式出现——如今,它出现在每本本科微积分教科书中,它将“线积分”与“曲面积分”联系起来。如今,许多学生发现它很难应用,更不用说证明。没有人知道麦克斯韦或他的任何同学是否也证明了这一点,但我想没有,因为麦克斯韦后来将完整的证明归功于威廉·汤姆森,即未来的开尔文勋爵。7

You can get a feel for the difficulty of the Smith’s Prize exams from question 8 on the February 1854 paper, which Maxwell sat: it asked for a proof of what is now known as “Stokes’s theorem”—the Irish-born mathematician George Stokes had been setting the Smith’s Prize papers for the past few years. The 1854 paper was the first time this theorem appeared in print—today it’s in every undergrad calculus textbook, and it relates a “line integral” to a “surface integral.” Many students today find it difficult enough to apply, let alone prove. No one knows if Maxwell, or any of his fellow examinees, proved it, either, but I suspect not, for Maxwell later credited the full proof to William Thomson, the future Lord Kelvin.7

线积分和曲面积分将在我们的故事中扮演重要角色,所以如果你以前没有遇到过它们,让我给你介绍一下它们是什么。我们在学校学到的普通积分学是对独立(水平)变量(通常是xt)求函数积分——所以当你求积分时,你是在把曲线下所有细长矩形的面积相加,就像我们在图 2.1中看到的那样。相比之下,一个简单的“线积分”例子是图 2.3b,其中我对一条线的一小段ds(在这种情况下是圆)求积分以求出圆周长。换句话说,在“线积分”中,你要把无穷小数量的小长度而不是面积相加——所以它是艾哈迈斯和阿基米德数千年前所做的圆周近似值的精确版本。“曲面积分”对整个曲面做类似的事情——但它需要“二重积分”这样才能在二维中进行积分。例如,在图 2.3a中,我可以使用曲面积分更轻松地找到圆的面积,如您在尾注中看到的那样。8这里的关键点是您不仅可以对水平轴进行积分,还可以沿曲线和表面进行积分——如果使用三重积分,还可以对体积进行积分。

Line and surface integrals will play a role in our story, so in case you haven’t met them before, let me give you an idea of what they are. The ordinary integral calculus we learn at school integrates a function with respect to the independent (horizontal) variable, often x or t—so when you integrate, you’re adding up the areas of all those skinny rectangles under the curve, as we saw in figure 2.1. By contrast, a simple example of a “line integral” is figure 2.3b, where I integrated with respect to a small segment ds of a line—a circle in that case—to find the circumference. In other words, in a “line integral” you’re adding up an infinitesimal number of little lengths rather than areas—so it’s an exact version of the circumference approximations Ahmes and Archimedes made thousands of years ago. A “surface integral” does a similar thing over a whole surface—but it needs a “double integral” so that you can integrate in two dimensions. For example, in figure 2.3a I could have found the area of the circle more easily using a surface integral, as you can see in the endnote.8 But the crucial point here is the idea that you can integrate not just with respect to the horizontal axis, but also along curved lines and across surfaces—and through volumes, too, if you use a triple integral.

至于麦克斯韦,十年后,他将充分利用他在史密斯奖考试中第一次遇到的定理,但与此同时,尽管他的考试成绩优异,他却没有获得剑桥奖学金。显然,当权者认为他太粗心了——他公开承认了数学上的缺陷,我相信我们很多人都能感同身受。因此,才华横溢但有时草率的麦克斯韦毕业了,却没有工作,而几个月后,泰特被提升为贝尔法斯特皇后学院的数学教授;他的《四元数》 仍未完成,等待他抽出时间去研究。这表明麦克斯韦的困境也有好的一面,因为他的父亲生活宽裕:在剑桥担任私人教师和在新成立的工人学院做志愿者的间隙,他有时间在家乡格伦莱尔静静地思考迅速发展的电磁学这一新科学——这是一座美丽的庄园,至今仍与麦克斯韦时代的样子一模一样(这在很大程度上要归功于其现任主人邓肯·弗格森9 的努力)。

As for Maxwell, in ten years’ time he would make good use of the theorem he first met in the Smith’s Prize exam, but in the meantime, and despite his outstanding exam results, he wasn’t awarded a Cambridge fellowship. Apparently, the powers-that-be thought he was too careless—a mathematical failing he openly acknowledged, and one I’m sure many of us can identify with. So, the brilliant if sometimes slapdash Maxwell graduated without a job, while a few months later Tait was promoted to professor of mathematics at Queens College, Belfast; his copy of Quaternions still lay unfinished, waiting for him to find the time to study it. Which suggests there was an upside to Maxwell’s predicament, given that his father was comfortably off: between stints as a private tutor at Cambridge and volunteering at the newly founded Working Men’s College, he had time to contemplate the rapidly developing new science of electromagnetism, at home in the peace of Glenlair—a beautiful estate that still looks much as it did in Maxwell’s time (thanks largely to the efforts of its present owner, Duncan Ferguson9).

重获自由后,麦克斯韦做的第一件事就是给威廉·汤姆森写信,汤姆森在电学数学方面的工作已经给他留下了深刻印象。麦克斯韦说,他想尽快了解电磁研究的现状,并询问汤姆森能否建议他从哪里开始。

One of the first things Maxwell did in his newfound freedom was write to William Thomson, whose work on the maths of electricity had already made an impression on him. Maxwell said that he wanted to get himself up to speed on the state of electromagnetic research, and he asked if Thomson would advise him on where he should start.

这种大胆的做法并非完全出乎意料,因为他和汤姆森几年前就见过面。那是在 1850 年英国协会在爱丁堡举行的一次会议上,当时 19 岁的麦克斯韦——他仍然很害羞和笨拙,说话带有地方口音,说话犹豫不决——在提问时间勇敢地站起来,就刚刚提交给会议的一篇关于光学和视觉的论文发表了看法。(麦克斯韦也是视觉和色彩研究的先驱,他后来研究如何拍摄出有史以来第一张日常物品的彩色照片——一条格子缎带,他是一位自豪的苏格兰人。)据一位与会者说,观众“半是困惑,半是焦虑,也许还有点怀疑……他们注视着这个看起来粗鲁的年轻人,他用断断续续的口音向他们讲话。”因此,你可以想象,当会议结束后,数学物理学界的一颗新星——26 岁的汤姆森找到麦克斯韦,想进一步了解他的想法时,麦克斯韦是多么惊讶和欣慰。10

This bold approach was not entirely out of the blue, for he and Thomson had met several years earlier. It was at an Edinburgh meeting of the British Association in 1850, where nineteen-year-old Maxwell—still shy and awkward, with his regional accent and hesitant way of speaking—had bravely stood up to make a point at question time, about a paper on optics and vision that had just been presented to the meeting. (Maxwell was also a pioneer of the study of vision and colour, and he would work out how to create the first-ever colour photograph of an everyday object—a tartan ribbon, proud Scot that he was.) According to one attendee, the audience was “half-puzzled, half-anxious and perhaps somewhat incredulous … as they gazed on the raw-looking young man who, in broken accents, was addressing them.” So, you can imagine Maxwell’s surprise and relief when, after the meeting had finished, he was approached by a rising star in the world of mathematical physics: twenty-six-year-old Thomson, who wanted to know more about his ideas.10

1854 年,汤姆森慷慨地回复了麦克斯韦的信,在他的建议下,麦克斯韦开始仔细阅读迄今为止已知的所有电磁结果。他本可以在大学学习一些相关的数学和物理学,但在接下来的几页中,我将带你简要回顾一下他需要了解的关键历史知识。在改造这个主题之前,麦克斯韦必须先了解所有可能,从而在 Tait 的帮助下开创了一种全新的物理学方法:矢量场。或者更确切地说,麦克斯韦采用了其他人直观地瞥见的方法,并为其注入了新的活力,使它真正成为一种全新的方法。

Now, in 1854, Thomson replied generously to Maxwell’s letter, and with his advice, Maxwell set himself the task of reading up carefully on all the electromagnetic results known so far. He would have studied some of the relevant maths and physics at university, but in the next few pages I’ll take you on a brief historical journey through the key things he needed to know before he could transform the subject—and thereby inaugurate, with Tait’s help, a brand-new approach to physics: the vector field. Or rather, Maxwell would take an approach that others had intuitively glimpsed and breathe such new life into it that it was, indeed, brand-new.

图像

麦克斯韦和他的彩色陀螺。剑桥大学三一学院,Add.P.270a;经剑桥大学三一学院院长和研究员友情许可。

Maxwell and his colour top. Cambridge, Trinity College, Add.P.270a; by kind permission of the Master and Fellows of Trinity College, Cambridge.

麦克斯韦的 1854 年之旅——以及过多的积分

MAXWELL’S JOURNEY IN 1854—AND A SURFEIT OF INTEGRALS

汉斯·奥斯特 (Hans Øersted) 于 1820 年发现电流的磁效应,但在 1800 年亚历山德罗·伏特 (Alessandro Volta) 发明第一块电池之前,人们不可能观察到这一现象。在此之前,只能将静电储存在一种早期的电容器中,即莱顿瓶。静电通常是由摩擦产生的,比如在黑暗的房间里梳头时,会看到电火花。梳头会将电子从头发的原子中分离出来——尽管当时没有人知道电子。但通常两种材料只需相互接触即可交换电子,就像麦克斯韦所说的“接吻的电”。 (与此相关,大多数手机屏幕中都嵌入了电容器(电荷源),因此当您触摸屏幕时,电子会在屏幕和手指之间流动:由此产生的电容变化被解释为触摸命令。)

Hans Øersted’s 1820 discovery of the magnetic effect of an electric current could not have been observed before 1800, when Alessandro Volta invented the first electric battery. Before then, only static electricity could be stored, in an early form of a capacitor called a Leyden jar. Static electricity is often generated by friction, as when you brush your hair in a darkened room and see sparks of electricity. The brushing knocks electrons out of the atoms in your hair—although no one knew about electrons back then. But often it is enough for two materials just to touch each other for electrons to be exchanged—as in what Maxwell called “the electricity of kissing.” (In a related vein, capacitors—sources of charge—are embedded in most mobile phone screens, so that when you touch the screen electrons flow between it and your finger: the resulting change in capacitance is interpreted as a touch command.)

因此,电力的第一批数学研究是关于静电荷的影响——电荷倾向于停留在原地而不是在电流中自由移动。特别是,1785 年,查尔斯·奥古斯丁·库仑用扭力秤做了他著名的实验,并表明带电粒子的电力与离源距离的平方成反比——就像牛顿引力定律中静止物体的质量施加的力一样。令人惊讶的巧合是,两个如此不同的现象的数学定律竟然具有相同的形式。11

So, the first mathematical studies of electricity were about the effects of static charges—charges that tend to stay put rather than freely moving in a current. In particular, in 1785 Charles Augustin Coulomb did his famous experiment with a torsion balance and showed that the electric force from a charged particle drops off in inverse-square proportion to the distance from the source—just like Newton’s law of gravity for the force exerted by a stationary object’s mass. It was an astonishing coincidence that the mathematical laws of two such different phenomena should have the same form.11

大约在这个时候,约瑟夫-路易斯·拉格朗日将他强大的数学头脑运用到引力理论的新公式中——就像平方反比定律一样,它很快也被用于描述电学。拉格朗日的公式涉及数学“势”的概念——一个与“势能”和力所做的“功”有关的概念。例如,要克服重力举起重物需要努力——做功,但一旦重物被举起来,如果重物掉下来,就有可能消耗能量做更多的功。你可以在起重机上吊起的巨大爆破球或驱动涡轮机的落水中看到这种应用。然而,要举起爆破球,你需要对它施加力,就像牛顿第二运动定律一样;这种力所做的“功”定义为“力乘以物体移动的距离”。在尾注中,我给出了“功”的数学原理以及它如何得出势能——但主要观点是,功的定义涉及积分。12

At around this time, Joseph-Louis Lagrange brought his powerful mathematical mind to bear on a new formulation of the theory of gravity—and just like the inverse square law, it would soon prove useful in describing electricity, too. Lagrange’s formulation involved the concept of a mathematical “potential”—an idea related to “potential energy” and the “work” done by a force. For instance, it takes effort—work—to lift a weight against gravity, but once it’s up there it has the potential to expend energy doing more work if it falls. You can see this kind of application in the huge demolition balls hoisted on cranes or in the falling water that drives a turbine. To lift up the demolition ball, though, you need to apply a force to it, à la Newton’s second law of motion; the “work” done by this force is defined as “force times the distance the object is moved.” In the endnote, I’ve given the maths of “work” and how it leads to the potential—but the main point is that the definition of work involves an integral.12

如今,“电压”(以伏特命名)可能更适合用来描述电势,它的作用类似于热量中的温度和流体中的压力。如果一根装满水的管道两端存在压力差,那么水就会从高压端流出端到低压端。同样,热量从较热的区域流向较冷的区域。当事物发生这样的变化时,导数就会发挥作用,因此,您不会惊讶地发现,如果V是与力相关的势,那么力的分量是:

Today, “voltage” (named in honour of Volta) might be a more familiar term for electric potential, and its role is similar to that of temperature in heat and pressure in fluids. If there’s a pressure difference between the ends of a pipe filled with water, then the water flows from the high-pressure end to the low-pressure end. Similarly heat flows from a hotter region to a colder one. When things change like this, derivatives come into play, so you won’t be surprised to find that if V is the potential associated with a force, then the force’s components are:

Vx,Vy,Vz.

大写的 d 表示“偏”导数,表示V在整个空间中变化,而不仅仅是像我们熟悉的dd. 13

The curly d’s indicate “partial” derivatives, which show that V is changing all through space, not just in one direction as with the familiar dydx.13

拉格朗日思想在电学和磁学中的应用是几位数学家几十年工作的成果,首先是皮埃尔-西蒙·拉普拉斯,他比拉格朗日更充分地发展了势理论。然后是无处不在、多才多艺的卡尔·弗里德里希·高斯,了不起的自学成才的数学家乔治·格林(当时他的日常工作是当磨坊主,尽管十年后,43 岁的他以第四名的成绩从剑桥大学毕业)以及拉普拉斯和拉格朗日的学生西蒙-丹尼斯·泊松。1817 年,玛丽·萨默维尔和她的丈夫访问巴黎时,拉普拉斯邀请他们到他的乡间别墅做客——她是英国为数不多的研究并理解他的《天体力学》的人之一。她发现拉普拉斯非常善良、细心,而泊松是一位有趣而活泼的晚宴客人。

The application of Lagrange’s idea to electricity—and to magnetism, too—was the result of the work of several mathematicians over several decades, beginning with Pierre-Simon Laplace, who developed potential theory more fully than Lagrange. Then came the ubiquitous and versatile Carl Friedrich Gauss, the remarkable self-taught mathematician George Green—whose day job at the time was being a miller, although a decade later, at the age of forty-three, he graduated from Cambridge as Fourth Wrangler—and Laplace and Lagrange’s student Siméon-Dénis Poisson. When Mary Somerville and her husband visited Paris in 1817, Laplace invited them to his country house—she was one of the few in Britain who had studied and understood his Mécanique Celeste. She found Laplace very kind and attentive, and Poisson an entertaining and vivacious dinner guest.

然而,索菲·热尔曼与泊松的经历却截然不同,因为她竟敢与他在同一领域竞争。她以匿名形式在法国科学院主办的征文比赛中提交了一篇关于振动表面理论的开创性论文——这是一个通过物理发现(当沙子散落在因声音而振动的盘子上时形成的图案)为数学开辟的全新主题。这是一个非常困难的问题,就像格拉斯曼参加莱布尼茨百年纪念竞赛一样,热尔曼的论文是唯一提交的论文。她的身份很​​可能是一个公开的秘密——但她的自学能力确实显露出来,她的推导并不完全正确,所以她的初稿被拒绝了。她重新修改了论文并再次提交——只有标准足够高才会获奖;除此之外,竞赛仍未结束。也许是因为她是局外人,作为评委之一的泊松表现得相当糟糕,严厉批评了她的第二篇论文(其他评委给了它荣誉奖),然后拿起结论自己研究。她没有气馁,继续改进论文,最终在 1816 年第三次尝试时赢得了奖项。我在这里提到它,不仅仅是为了赞扬杰曼和她不顾女性在科学领域面临的障碍而坚持不懈,也因为振动表面的数学包含一些与大约同一时期为重力、电力和磁力开发的方程和表达式类似的方程和表达式。

Sophie Germain, however, had a rather different experience with Poisson, for she had the temerity to compete in the same playing field as he. She anonymously entered a pioneering paper on the theory of vibrating surfaces—a brand-new topic opened up to mathematics through a physical discovery (the patterns made when sand is scattered on a plate set vibrating by a sound)—in an essay competition sponsored by the French Academy of Sciences. It was such a difficult problem that, just as with Grassmann’s entry in the Leibniz centenary competition, Germain’s was the only essay submitted. It is likely her identity was an open secret—but it’s true that her self-education showed, and her derivations weren’t completely correct, and so her first effort was rejected. She reworked her paper and submitted it again—prizes were only awarded if the standard was high enough; otherwise, the competition remained open. Perhaps because she was an outsider, Poisson, who was one of the judges, behaved rather badly, by severely criticising her second essay (the other judges gave it an honourable mention) and then lifting its conclusion and working on it himself. Undeterred, she kept on improving her paper and finally won the prize with her third attempt, in 1816. I mention it here not just as a shout out to Germain and her persistence despite the obstacles for women in science, but also because the maths of vibrating surfaces includes some similar equations and expressions to those being developed for gravity, electricity, and magnetism around the same time.

例如,拉普拉斯已经证明势能V遵循一个非常简单的方程:

For instance, Laplace had shown that the potential V obeyed a really neat equation:

22+22+22=0.

2Vx2+2Vy2+2Vz2=0.

泊松后来证明,在物理应用中,右边并不总是零——这取决于你测量的内容和位置。无论哪种情况,麦克斯韦都将左边表达式的物理意义描述为势的“浓度”,在数学上,它今天被称为V的“拉普拉斯算子” ,其中拉普拉斯算子是“微分算子”

Poisson later showed that in physical applications the right-hand side is not always zero—it depends on what and where you’re measuring. Either way, Maxwell described the physical meaning of the expression on the lefthand side as the “concentration” of the potential, and mathematically it is known today as the “Laplacian” of V, where the Laplacian is the “differential operator”

22+22+22

2x2+2y2+2z2

(或者麦克斯韦所说的“拉普拉斯算子”)。

(or “Laplace’s operator,” as Maxwell called it).

正如我在第二章中提到的dd中,“算符”一词仅表示表达式指示您“操作”尚未插入的某个函数(在本例中为求其二阶(偏)导数)。在物理学中,您要操作的函数具有物理意义,例如势能,但从数学上讲,任何适当可微的函数都可以。当我们讨论张量时,函数“插入”时“操作”的思想将变得尤为重要。

As I mentioned in chapter 2 in connection with ddx, the term “operator” just means that the expression directs you to “operate on”—in this case, take the second (partial) derivatives of—some function that is yet to be inserted. In physics, the functions you want to operate on have a physical meaning, such as the potential, but mathematically speaking, any suitably differentiable function will do. The idea of something that “operates” when a function is “inserted” will become especially important when we come to tensors.

与此同时,自然与数学似乎以一种奇妙的方式交织在一起,拉普拉斯算子不仅出现在势理论中,还出现在热、振动和波的理论中:除了使用势V,还可以取温度T的拉普拉斯算子,例如在“热方程”中,或者取波的拉伸距离u,例如在“波动方程”中。然而,这对我们的故事的意义在于,拉格朗日、拉普拉斯和其他先驱者只是使用了笛卡尔坐标和分量。他们有牛顿的矢量概念,但全矢量微积分尚未发明。

Meantime, in the wonderful way that nature and mathematics seem intertwined, the Laplacian appears not only in potential theory but in the theory of heat, vibrations, and waves: instead of using the potential V, you can take the Laplacian of the temperature T, say, as in the “heat equation,” or the stretching distance u in a wave, as in the “wave equation.” The significance of this for our story, though, is that Lagrange, Laplace, and the other pioneers simply used Cartesian coordinates and components. They had the Newtonian idea of a vector, but whole-vector calculus had yet to be invented.

然而,一旦 Tait 静下心来阅读 Hamilton 的四元数,他就会开始以全矢量形式应用势能和拉普拉斯算子——然后 Maxwell 会利用这些矢量量获得前所未有的优势。但那还是 15 年后的事。1854 年,Maxwell 仍在阅读他进入拳击场之前的内容,而 Tait 仍在忙于他在贝尔法斯特的新学术工作。

Once Tait settled down to reading Hamilton’s Quaternions, however, he would begin to apply the potential and Laplacian in full vector form— and Maxwell would then use these vector quantities to unprecedented advantage. But that was still fifteen years in the future. In 1854, Maxwell was still reading up on what came before he entered the ring, and Tait was still busy with his new academic duties in Belfast.

• • •

• • •

好像所有这些新的静电计算还不够具有挑战性,当奥斯特发现电磁学的存在(涉及在流动的电流中移动带电粒子)时,事情变得更加复杂。

As if all this new calculus of static electricity wasn’t challenging enough, when Øersted discovered the existence of electromagnetism—which involved moving charged particles in a flowing electric current—things became a whole lot more complicated.

1821 年,安德烈-马里·安培利用 Øersted 的奇特发现开创了电报技术——他还开始尝试通过实验量化电磁效应,然后用数学表达它们。他取得了惊人的进步,独自开创了电磁学(或他称之为“电动力学”)的数学研究。麦克斯韦称他为“电学界的牛顿”,因为他是一位伟大的实验家数学理论家。特别是,他使用线积分和面积积分来量化电流和它在各种实验装置中产生的磁力之间的关系——但当他于 1836 年去世时,他仍然没有找到对电磁学的完整通用描述。14

In 1821, André-Marie Ampère used Øersted’s strange discovery to pioneer telegraphy—and he also began trying to quantify electromagnetic effects experimentally and then express them mathematically. He made amazing progress, single-handedly initiating the mathematical study of electromagnetism (or as he called it, “electrodynamics”). Maxwell dubbed him the “Newton of electricity,” because he was such a great experimenter and mathematical theorist. In particular, he used line and surface integrals to quantify the relationship between electric current and the magnetic force it produces in various experimental set-ups—but when he died in 1836, he still hadn’t managed to find a completely general description of electromagnetism.14

在安培开始电磁研究的同时,英国人迈克尔·法拉第利用奥斯特的发现开发了一种电动机的原型。然后,在 1831 年,也就是麦克斯韦和泰特出生的那一年,法拉第发现了与 Øersted 的相反的效应:他证明了可以通过移动磁铁穿过线圈来产生电流。(正是相对运动产生了电流,所以你也可以转动线圈而不是磁铁,我们很快就会在图 6.2中看到这种装置。)几个月后,美国人约瑟夫·亨利 (Joseph Henry) 发现了同样的现象,这又是一个科学上独立共同发现的例子。

At the same time as Ampère began his electromagnetic research, Englishman Michael Faraday used Øersted’s discovery to develop a rudimentary prototype of an electric motor. Then, in 1831, the year Maxwell and Tait were born, Faraday discovered the inverse effect to Øersted’s: he showed that you could generate an electric current by moving a magnet through a coil of wire. (It is the relative motion that induces the current, so you can just as well turn the coil instead of the magnet, and we’ll see this set-up shortly, in fig. 6.2.) In yet another example of independent codiscovery in science, the American Joseph Henry discovered the same thing, some months later.

法拉第用一台原型发电机展示了这一非凡现象,后来其他人将其发展成为商业产品。剩下的就成了历史,无论好坏,因为到了 19 世纪后期,世界开始通电。例如,在 19 世纪 80 年代,汤姆森建造了一座房子,这是世界上最早使用电灯的房子之一;而在德国和意大利,爱因斯坦的父亲和叔叔经营着一家创新型科技公司,制造和安装发电机,首次将电力带入企业、公共建筑、街道甚至家庭。当然,今天我们很遗憾大多数大型发电机都是燃煤的——在没有水流或风等自然力的情况下,需要热量来产生蒸汽,才能转动产生电流的巨大线圈。尽管如此,我们中很少有人愿意没有电,所以我们不仅要感谢设计出可持续、气候安全的电力生产方式的现代科学家和工程师,还要感谢试图弄清楚电力如何运作的十九世纪实验学家和数学家。

Faraday demonstrated this remarkable phenomenon with a prototype generator, which others would later develop into commercial propositions. The rest is history, for better and worse, for by the late 1800s the world began to go electric. In the 1880s, for example, Thomson built a house that was one of the first in the world to include electric light, while in Germany and Italy, Einstein’s father and uncle were running an innovative technological company making and installing generators, bringing electricity to businesses, public buildings, streets, and even households for the very first time. Today, of course, we rue the fact that most large generators have been coal-fired—in the absence of a natural force such as falling water or wind, you needed heat to produce steam, in order to turn the huge coils that generate the current. Still, few of us would want to do without electricity, so we owe thanks not just to the modern scientists and engineers designing sustainable, climate-safe ways of producing it, but also to the nineteenth-century experimentalists and mathematicians who were trying to figure out just how it all works.

当麦克斯韦首次踏上电磁之旅时,理清其中的数学问题变得相当棘手。这与在科学中使用类比的优势和陷阱有关——在本例中,类比是指重力、流体动力学和电力之间的类比。我已经提到过重力和静电之间的数学相似性,但流体动力学——研究流体在力的作用下如何表现的学科——之所以重要,是因为在早期,人们倾向于将电视为流体。这就是为什么他们用“电流”这个词来描述“带电流体”的流动,不管它是什么:电子直到十九世纪末才被发现。但麦克斯韦意识到,很少有人意识到这一点,类比可能会让你误入歧途——事实上,电根本不是一种流体。不过,他知道,至少在最初,这种类比为理解电磁学这一新现象的数学提供了一种及时而有力的方法。挑战在于使用数学而不将其与物理现实混淆。

When Maxwell first set out on his electromagnetic journey, sorting out the maths of it had become rather tricky. It had to do with the advantages and pitfalls of using analogies in science—in this case, between gravity, hydrodynamics, and electricity. I’ve mentioned already the mathematical parallels between gravity and static electricity, but hydrodynamics—the study of how fluids behave as a result of forces—was important because in those early days people tended to visualise electricity as a fluid. That’s why they used the word “current” to describe the flow of “charged fluid,” whatever it was: the electron wouldn’t be discovered until the very end of the nineteenth century. But Maxwell realised, as very few had done, that analogies could lead you astray—indeed, electricity turned out not to be a fluid at all. Still, he knew that initially, at least, this analogy had offered a timely and powerful way into the mathematics of the new phenomenon of electromagnetism. The challenge was to use the maths without confusing it with the physical reality.

例如,另一个与流体有明显联系的术语是“通量”。牛津英语词典将其定义为“流动或流出的过程”。但在 18 世纪和 19 世纪初的流体研究中,“通量”获得了更具体的含义,即在给定时间内流过给定表面(例如管道的横截面积)的流体量。你可以把它想象成每秒计算离开管道的流体分子数,你可以用质量或体积来测量这个数字。在图 6.1中,我用体积显示了通量。

For instance, another term that has evident links with fluids is “flux.” The Oxford English Dictionary defines it as “a process of flowing or flowing out.” But in the study of fluids in the eighteenth and early nineteenth centuries, “flux” acquired a more specific meaning, as the amount of fluid flowing through a given surface—such as the cross-sectional area of a pipe—in a given time. You can think of it as counting the number of molecules of fluid leaving the pipe each second—and you can measure this number using mass or volume. In figure 6.1, I’ve shown the flux using volume.

一旦你做出了这样的定量定义,你就可以对其进行数学定义。对于流过圆柱形管道的流体,通量就是管道横截面积和流体速度的标量积(如图6.1 的标题所示)。对于流过任何形状表面的通量,而不仅仅是具有简洁面积公式的圆,其定义涉及表面积分。这是因为你实际上是在将流过表面上每个小面积元素的液体量相加(积分)。

Once you make a quantitative definition like this, you can define it mathematically. For fluid flowing through a cylindrical pipe, the flux turns out to be the scalar product of the cross-sectional area of the pipe and the velocity of the fluid (as you can see in the caption to fig. 6.1). In the case of flux through a surface of any shape, not just a circle with its neat area formula, the definition involves a surface integral. That’s because you’re effectively adding up (integrating) the amount of liquid flowing across each little element of area on the surface.

所有这些都表明,通量是任何矢量的属性,而不仅仅是流体的速度。果然,正如库仑静电定律和牛顿引力定律具有相同的数学形式,正如拉格朗日势能和拉普拉斯-泊松方程可以应用于电和重力一样,事实证明,通量积分可以应用于流体以外的其他事物——包括重力、电和磁。高斯是发展这种数学的先驱之一,因此将通过封闭表面的引力和电通量分别与封闭的质量和电荷量联系起来的定律被称为“高斯定律”。事实上,你可以从高斯定律推导出牛顿定律和库仑定律——而且磁通量也有高斯定律。不过,这里的重点是,所有这些早期的通量应用都涉及积分。但物理上,真正有趣的事情发生在这些电和磁通量发生变化,因为此时电磁效应开始发挥作用。因此,这些变化的磁通量是电磁学实际应用的关键,如图 6.2中电动机和发电机的示意图所示。

All this suggests that flux is a property of any vector, not just the velocity of a fluid. And sure enough, just as Coulomb’s law of static electricity and Newton’s law of gravity have the same mathematical form, and just as Lagrange’s potential and the Laplace-Poisson equations can be applied to both electricity and gravity, so it turned out that flux integrals could be applied to other things than fluids—including gravity, electricity, and magnetism. Gauss was one of the pioneers in developing this maths, so the laws relating the gravitational and electrical flux through a closed surface to the enclosed amount of mass and charge, respectively, are called “Gauss’s laws.” In fact, you can deduce Newton’s and Coulomb’s laws from Gauss’s laws—and there’s a Gauss’s law for magnetic flux, too. The main point here, though, is that all these early applications of flux involved integrals. But physically, the really interesting things happen when these electric and magnetic fluxes change, for that’s when electromagnetic effects come into play. So, these changing fluxes hold the key to practical applications of electromagnetism, as in the schematic setup for electric motors and generators in figure 6.2.

图像

图 6.1。流过管道的流体通量。此处流体的速度(箭头所示)是恒定的。其方向垂直于(或“法向”)管道末端所示的表面,因此在这种情况下,所有水都流过该表面。我在中间突出显示的小圆柱形流体部分的体积为V = Ad,即圆形底部的面积为A(= π r 2)乘以长度d。因此,流体以速率(单位时间的体积)流过管道末端的阴影表面一个d。 但d是距离/时间,即速度,因此速度矢量是所示方向上的速度。这意味着在给定时间内流过管道末端的流体量(即通量)为A v。这个公式很有意义,因为流速越快,面积越大,在给定时间内流过管道末端的流体体积(或分子数量)就越多。

FIGURE 6.1. Flux of fluid flowing through a pipe. Here the velocity of the fluid (shown by the arrows) is constant. Its direction is perpendicular (or “normal”) to the surface shown at the end of the pipe, so in this case all the water flows through that surface. The small cylindrical section of fluid that I’ve highlighted in the middle has volume V = Ad, i.e., the circular base with area A (= πr2) times the length d. So, the fluid flows through the shaded surface at the end of the pipe at a rate (volume per unit time) of Adt. But dt is distance/time, which is the speed, so the velocity vector is this speed in the direction shown. Which means the amount of fluid flowing through the end of the pipe in a given time—that is, the flux—is Av. This formula makes sense, because the faster the flow and the larger the area, the more volume (or number of molecules) of fluid passes through the end of the pipe in a given time.

当管道出口部分堵塞或与流动方向成一定角度时,情况会更加复杂。例如,如果流动方向与流经表面(法线)成 θ 角,则需要向量v在法线方向上的分量。这是因为通量是流经(即穿过)表面的流体量。因此,通量的公式变为标量积,Av (= Av cos θ)。将A设为矢量意味着指定表面的方向定义为其法线的方向。但是,当表面弯曲或形状不规则时,您需要使用向量微积分来求面积。然后,一般通量公式为sd一个。但我提到这一点只是为了完整性——我们不会再需要它了。

It’s more complicated when the pipe outlet is partially blocked, or at an angle to the flow. For instance, if the flow direction is at an angle θ to (the normal to) the surface it’s flowing through, you need the component of the vector v in that normal direction. That’s because flux is the amount of fluid flowing through—that is, across—the surface. So, the formula for flux becomes the scalar product, Av (= Av cos θ). Making A a vector means specifying that the direction of a surface is defined to be that of its normal. When your surface is curved or irregularly shaped, however, you need to use vector calculus to find the area. Then the general flux formula is svdA. But I’m mentioning this just for completeness—we won’t need it again.

许多研究人员都在试图弄清这一现象究竟是如何发生的,以及如何用数学表达出来,其中包括法国的安培和奥古斯丁·柯西、英国的格林和汤姆森以及德国物理学家威廉·韦伯。麦克斯韦阅读了他们的所有著作,并钦佩他们的各种发现和假设。然而,这些杰出人物中没有一个人能够找到完整的电磁理论。

Exactly how this happened, and how it could be expressed mathematically, is what a host of researchers were trying to find out, including Ampère and Augustin Cauchy in France, Green and Thomson in Britain, and the German physicist Wilhelm Weber. Maxwell read up on them all and admired their various discoveries and hypotheses. Yet none of these luminaries had managed to find a complete theory of electromagnetism.

图像

图 6.2电动机和发电机:正如 Øersted 所发现的,电流可以产生磁力,而人们早就知道两个磁铁会相互吸引或排斥(异极吸引)。因此,当将载流线圈放置在外部磁“场”中时(此处用磁极之间的箭头表示),其自身的磁场会对外部磁场作出反应。这会使线圈偏转并转动 - 电动机可以利用这种转动效果(扭矩)。当线圈转动时,它与外部磁场的夹角会发生变化 - 这意味着穿过线圈的磁通量也会发生变化(参见图 6.1)。因此,如果先机械地转动线圈,那么穿过线圈的变化的磁通量就会感应出产生电流所需的电磁场 - 这就是发电机的基础。

FIGURE 6.2. Electric motors and generators: As Øersted discovered, an electric current can produce a magnetic force, and it had long been known that two magnets attract or repel each other (opposite poles attract). So, when a current-carrying coil is placed in an external magnetic “field”—shown here by the arrows between the magnetic poles—its own magnetic field reacts to the external one. This deflects the coil and turns it—and this turning effect (torque) is harnessed in electric motors. As the coil turns, the angle it makes with the external magnetic field changes—which means the magnetic flux through the coil is changing (cf. fig. 6.1). So, if you mechanically turn the loop first, then the changing magnetic flux through the loop induces the electromagnetic field needed to produce an electric current—and this is the basis of electricity generators.

现在我们开始看到麦克斯韦的哲学训练在哪里脱颖而出——更不用说他的天才了。虽然不同物理现象之间的类比当然非常有用,但它们带有隐藏的假设——麦克斯韦意识到,他的前辈选择的数学工具,包括所有这些积分,都体现了最大的假设。因为大多数研究人员都假设引力、电力、磁力和电磁力都是超距作用的——瞬间直接从一个物体跳跃到另一个物体,一个带电粒子或一个磁铁跳跃到另一个磁铁,以及从移动的磁铁跳跃到导线或从载流导线跳跃到磁铁。麦克斯韦是唯一一个清楚地认识到,我在这里谈论的通量和其他积分被认为是体现超距作用的人,因为它们只关注构成积分极限或积分边界的点、线和。例如,重力或电力F将物体从点r 1移动到点r 2 时所做的功为r1r2F×dr;进行这样的积分时,只需要极限积分的极限——你不需要知道r 1r 2之间的路径。15所以你可以看到,对于那些相信所有作用都发生在两点、从一个点到另一个瞬间相互作用的人来说,这种类型的积分是正确的工具。再举一个例子,你只需要在(现代版)牛顿证明中积分的极限,即两个球形物体(例如太阳和行星)之间的引力吸引力就好像它们所有的质量都集中在它们的中心一样。你可以明白为什么这表明了超距作用:就好像这两个中心点之间没有发生任何引力一样。麦克斯韦意识到,同样,他的前辈使用的所有电磁积分都没有考虑可能发生的任何过程,不仅仅是在边界,而且是中间或周围空间。也许这就是为什么没有人能够破解电磁密码的原因。

Now we begin to see where Maxwell’s training in philosophy comes to the fore—not to mention his genius. While analogies between different physical phenomena can certainly be very useful, they come with hidden assumptions—and Maxwell realised that his forerunners’ choice of mathematical tools, with all those integrals, embodied the biggest assumption of all. For most researchers assumed that gravity, electricity, magnetism, and electromagnetism all acted at a distance—instantaneously leaping directly from one material body to another, one charged particle, or one magnet, to another, as well as from a moving magnet to a wire or from a current-carrying wire to a magnet. Maxwell was the only one who recognised clearly that the flux and other integrals I’ve been talking about here were assumed to embody action-at-a-distance, because they focussed only on the points, lines, and surfaces that formed the limits of integration or boundaries of the integral. For instance, the work done by a gravitational or electric force F in moving an object from point r1 to point r2 is given by r1r2F×dr; when you carry out such an integral, you need only the limits of integration—you don’t need to know anything about the path between r1 and r2.15 So you can see that this type of integral was the right tool for those who believed all the action happened at the two points, interacting instantaneously from one point to the other. To take another example, you only need the limits of integration in (the modern version of ) Newton’s proof that the gravitational attraction between two spherical objects, such as the Sun and a planet, acts as if all their mass is concentrated at their centres. You can see why this suggested action at a distance: it’s as if nothing gravitational happened in between these two central points. Maxwell realised that, similarly, all those electromagnetic integrals his predecessors were using took no account of any processes that might be going on not just at the boundaries, but also within the intervening or surrounding space. Perhaps that was why no one had managed to crack the electromagnetic code.

但这一切即将改变。经过一年的认真研究,麦克斯韦准备抛开主流数学家的著作,接受一个完全外行人提出的想法。

But all this was about to change. After his year of careful study, Maxwell was ready to set aside the works of the mainstream mathematicians, and to take up an idea suggested by a rank outsider.

绝妙的想法

A BRILLIANT IDEA

法拉第早在 1831 年就通过实验发现了电磁感应,那一年麦克斯韦出生。然而,就数学理论而言,自学成才的法拉第无法理解那些花哨的数学术语,如线积分、曲面积分、高斯通量定律等。他十三岁就辍学,在善良的装订师乔治·里博手下当学徒,里博在法国大革命期间逃离法国。法拉第刚到里博的工作室时几乎不会读写——他的家人属于一个基督教原教旨主义教派,他们对学习兴趣不大,因为圣经是他们的权威,他经常逃学,也经常遭到严厉的校长的惩罚。但里博鼓励年轻的法拉第在下班后多多练习,试着阅读他们装订的一些书籍——正是通过这种非凡而艰苦的方式,法拉第对电产生了浓厚的兴趣,这要归功于新装订的大英百科全书的条目。他建立了一个基本的他进入了里鲍商店后面的实验室,参加公开讲座,最终学到了足够的知识,给皇家研究院的领军人物汉弗莱·戴维留下了深刻的印象——从那时起,他从实验室清洁工做成了戴维的助手,直到最终成为英国电磁研究实践和实验方面的顶尖专家。

Faraday had made his experimental discovery of electromagnetic induction back in 1831, the year Maxwell was born. As far as mathematical theory went, however, the self-taught Faraday hadn’t been able to follow all that fancy mathematical talk of line and surface integrals, Gaussian flux laws, and so on. He’d left school at thirteen, taking up an apprenticeship with the kindly bookbinder George Riebau, who had fled France during the Revolution. Faraday could barely read and write when he first arrived at Riebau’s workshop—his family belonged to a fundamentalist Christian sect that had little interest in learning because the Bible was its authority, and he’d often played truant from school and its harsh schoolmaster. But Riebau encouraged young Faraday to practice after work by trying to read some of the books they were binding—and in this extraordinary, painstaking way, Faraday had become enchanted with electricity, thanks to an entry in the newly bound Encyclopaedia Britannica. He set up a rudimentary laboratory at the back of Riebau’s shop, attended public lectures, and eventually learned enough to impress the Royal Institution’s leading light, Humphry Davy—and from there he worked his way from lab sweeper to Davy’s assistant until, ultimately, he became Britain’s leading expert on the practical, experimental side of electromagnetic research.

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图 6.3A。铁屑围绕磁铁排列。牛顿·亨利·布莱克和哈维·N·戴维斯,《实用物理学》(纽约:麦克米伦出版社,1913 年)。维基共享资源,公共领域。

FIGURE 6.3A. Iron filings aligning around a magnet. Newton Henry Black and Harvey N. Davis, Practical Physics (New York: Macmillan, 1913). Wikimedia Commons, public domain.

因此,尽管法拉第不太懂数学,但他对电磁的物理性质的了解肯定比大多数人都要多——这促使他开发出一种独特的方法来研究电荷和磁铁之间空间中发生了什么。他第一次产生这个想法是因为他注意到了一件可能也会让你着迷的事情,如果你上过学校的科学课的话:铁屑整齐地排列成环状,围绕着条形磁铁的两极。这是因为每个小铁屑都会将自己的小极点朝向条形磁铁的极点,如图 6.3a所示——正如麦克斯韦后来解释的那样,铁屑线之所以形成,是因为每次添加新的铁屑时,它也必须与相邻的铁屑首尾相连,因为吸引磁力集中在每个小磁铁末端附近的极点上。16

So, while Faraday didn’t know much maths, he certainly knew more than most about the physical nature of electromagnetism—and this led him to develop a unique approach to the question of what was going on in the space between electric charges and magnets. He first got the idea when he noticed something that may well have fascinated you, too, if you took a school science class: the neat way that iron filings line up in loops around the poles of a bar magnet. That’s because each little filing orients its own tiny poles to those of the bar magnet, as in figure 6.3a—and as Maxwell later explained, lines of filings are formed because each time a new filing is added, it must also align itself end to end with its neighbor, for the attracting magnetic force is concentrated at the poles near the ends of each little magnet.16

传统的“超距作用”理论认为,如果只有一块铁屑(如图 6.3b所示,一小块“测试”铁屑),那么它和磁铁之间的力将瞬间相互作用。换句话说,磁力只会在磁铁和铁屑之间起作用而且会瞬间起作用。但法拉第推断,即使你把所有其他铁屑都拿走,磁铁的力仍然会像以前一样作用在所有相同的位置,即使你再也看不到它的影响。特别是,一定有力一直作用磁铁和“测试”铁屑之间的空间,就像所有铁屑都存在时一样。为了描述这个想法,法拉第提出了“力线”的概念,与铁屑线直接类比。就好像铁屑使磁铁发出的不可见的力显现出来,就像光使水印可见一样。

The traditional, “action-at-a-distance” way of looking at this was that if you had just one iron filing present—a single little “test” filing, as in figure 6.3b—then the force between it and the magnet would interact instantaneously. In other words, the magnetic force would act only between the magnet and the filing, and it would do so instantaneously. But Faraday reasoned that even when you’d taken all the other iron filings away, the force from the magnet was still acting in all the same places as it was before— even if you could no longer see its effects. In particular, there must be force acting all the way through the space between the magnet and the “test” filing—just as there’d been when all the filings were present. To describe this idea, Faraday came up with the concept of “lines of force,” in direct analogy with the lines of iron filings. It was as if the filings made manifest the invisible force emanating from the magnet, just as light makes a watermark visible.

铁屑围绕磁铁排列的方式让人想起一片麦田。当微风吹过麦田时,每根小麦秆都会因风力而弯曲。同样,法拉第提出,电力和磁力都是通过带电粒子和磁铁产生的“场”传递的——他宣称,它们不会远处起作用。这是一个简单而绝妙的想法。问题是,他没有知道如何将其转化为可预测的数学理论,以便通过进一步的实验进行检验。因此,尽管他凭借其无可否认的熟练的实验研究立即获得了主流赞誉,但他的电场和磁场概念却几乎被忽视了。

The way iron filings line up around a magnet brings to mind something like a field of wheat. When a breeze blows across the field, each little stalk bends in response to the force of the wind. Similarly, Faraday proposed that both electric and magnetic forces were conveyed through “fields” emanating from charged particles and magnets—they did not, he declared, act at a distance. It was a simple, utterly brilliant idea. Trouble was, he didn’t know how to turn it into a predictive, mathematical theory that could be tested against further experiments. So, while he’d gained immediate mainstream acclaim for his undeniably skillful experimental research, his concept of electric and magnetic fields was all but ignored.

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图 6.3B。使每个单独的磁芯对齐的力(每个磁芯就像一个微小的条形磁铁)。法拉第也对电做了同样的事情,在中心电荷周围的空间中的各个位置放置一个小的“测试”电荷,并观察力在每个位置的作用。通过这种方式,他建立了电场和磁场的图像。

FIGURE 6.3B. The forces that align each individual filing (each of which is like a tiny bar magnet). Faraday also did the same thing for electricity, placing a little electric “test” charge at various spots in the space around a central charge and noticing how the force acted in each place. In this way he built up a picture of the electric and magnetic fields.

麦克斯韦对法拉第场做了什么

WHAT MAXWELL DID WITH FARADAY’S FIELD

事实上,在麦克斯韦出现之前,威廉·汤姆森几乎是唯一一位尝试将一些数学知识融入法拉第思想的知名理论家。汤姆森在年仅 17 岁时就对静电效应和热流进行了数学比较,借鉴了约瑟夫·傅立叶对热的数学研究。因此,麦克斯韦开始了自己的电磁职业生涯,将汤姆森的方法扩展到流体领域,将河流中的流线与法拉第磁铁周围的力线以及电荷周围的力线进行了类比。

In fact, before Maxwell came on the scene, William Thomson was virtually the only established theorist to try to put some maths into Faraday’s idea. When he was just seventeen, Thomson had made a mathematical comparison between electrostatic effects and heat flow, adapting Joseph Fourier’s mathematical study of heat. So, Maxwell began his own electromagnetic career by extending Thomson’s approach to fluids, making the analogy between the streamlines in a river and Faraday’s lines of force around a magnet, and around an electric charge.

在早期对热和流体流动的研究中,傅立叶、牛顿和欧拉等人实际上已经直观地想到了场的概念,尽管这是一种隐含的方式。但他们确实认为流体的每个粒子都有速度(和温度),这类似于法拉第的想法,即在带电体或磁铁周围的空间中,每个点的假设单元测试电荷和磁屑都会受到电力或磁力的作用。实际上,法拉第对力线的理解比对场的理解更清晰,汤姆森是物理学中第一个定义这一概念的人:只要空间中的每一点都有确定的磁力,他就会在 1851 年说,你就有一个“磁力场”,或者更简单地说,“一个磁场”。当然,“电场”也是如此。17

In early studies of heat and fluid flow, the likes of Fourier, Newton, and Euler had, in fact, intuited the idea of a field, albeit it an implicit way. But they did have the idea that each particle of fluid has a velocity (and a temperature), which is similar to Faraday’s idea that there are electric or magnetic forces acting on hypothetical unit test charges and filings placed at each point in the space around an electrically charged body or a magnet. Actually, Faraday had a clearer idea about lines of force than he did about the field, and it was Thomson who first defined the concept in physics: whenever there is a definite magnetic force at every point in a space, he’d said in 1851, you have a “field of magnetic force,” or more simply, “a magnetic field.” And similarly, of course, for an “electric field.”17

麦克斯韦将他的论文(第一篇关于电的论文)命名为“论法拉第的力线”,并在 1855-56 年冬天将其分两部分提交给剑桥哲学学会。(1855 年,他终于获得了剑桥的奖学金!)论文发表后,他寄了一份给法拉第。64 岁的法拉第已经厌倦了主流对他的场理论的拒绝,因此你可以想象当他发现 24 岁的麦克斯韦终于将其付诸实践时,他有多么高兴。法拉第热情地回复了麦克斯韦,两人成为了朋友。

Maxwell called his paper—his first on electricity—“On Faraday’s Lines of Force,” and he presented it to the Cambridge Philosophical Society in two parts over the winter of 1855–56. (In 1855 he’d finally secured a fellowship at Cambridge!) When it was later published, he sent a copy to Faraday. At the age of sixty-four, Faraday had grown weary of the mainstream’s rejection of his field idea, so you can imagine his joy on finding that twenty-four-year-old Maxwell had brought it to life at last. Faraday replied warmly, and the two men became friends.

当时,麦克斯韦还在照顾他深爱的父亲,他的父亲于 1856 年 4 月去世。他将自己的悲痛写进一首诗,然后尽自己所能,继续担任格伦莱尔的新领主,履行所有相关职责。(当然,这也是他的乐趣所在:麦克斯韦一直与庄园的佃户相处融洽,他和父亲还为那些想要提高阅读能力的人设立了一个项目。)几周后,他终于获得了阿伯丁马歇尔学院的教授职位,击败了亚瑟·凯莱(凯莱在当了多年律师后正试图重返学术界)和泰特。18两年后,他与学院校长的女儿凯瑟琳·杜瓦结婚。和汉密尔顿的妻子海伦一样,凯瑟琳也非常虔诚;麦克斯韦和汉密尔顿一样,对精神有着更广泛的看法,尽管他们都是虔诚的基督徒。

Maxwell had also been nursing his adored father, who died in April 1856. He poured his grief into a poem and then carried on as best he could as the new Laird of Glenlair, with all its attendant duties. (And its pleasures: Maxwell had always got on well with the estate’s tenants, and he and his father had set up a program for those who wanted to improve their reading.) A few weeks later, he finally landed a professorial appointment, at Marischal College in Aberdeen, beating Arthur Cayley to the job— Cayley was trying to get back into academia after his years as a lawyer— and Tait, too.18 Two years later he married the college principal’s daughter, Katherine Dewar. Like Hamilton’s wife, Helen, Katherine was strictly religious; Maxwell, like Hamilton, had a broader view of spirituality, although they were both committed Christians.

在接下来的十年里,麦克斯韦不断完善他对法拉第场的概念,并发表了许多其他论文,直到最后,他准备发表他的电磁场理论。它包括我提到的势能和通量的概念——但它包含的积分相对较少。相反,它有很多导数和“偏微分方程”——涉及关于x、yz的导数的方程,比如我之前展示的拉普拉斯算子中势能的导数。

Many other papers followed in the next decade as Maxwell honed his conception of Faraday’s field until, at last, he was ready to publish his radical field theory of electromagnetism. It includes the ideas of potential and flux that I’ve mentioned—but it contains relatively few integrals. Instead, there are many derivatives and “partial differential equations”—equations that involve derivatives with respect to x, y, and z, such as the derivatives of the potential in the Laplacian I showed earlier.

当然,他并不是唯一一个使用微分方程的人——自牛顿以来,人们就一直在物理学中使用它们。只是他比任何人都更清楚何时使用每种微积分,以及为什么使用。他仔细解释了他对电磁学中导数的偏爱,并指出,虽然线积分和曲面积分只关注诸如两个相互作用粒子之间的距离和“这些物体中的电或电流”之类的东西——如果你假设电力和磁力在远处作用,那么这是可以的——但偏微分方程显示了物体周围整个空间变化。因此,他认为,这样的方程是表达变化的电力和磁力和通量产生电磁效应的方式的自然工具。场,无论它是由什么构成的,都通过空间介导这些变化——就像麦田介导像波浪一样穿过它的微风一样。19

He wasn’t the only one to use differential equations, of course—people had been using them in physics ever since Newton. It’s just that he was clearer than anyone else about when to use each kind of calculus, and why. He explained his preference for derivatives in electromagnetism carefully, noting that while line and surface integrals focus only on such things as the distance between the two interacting particles and the “electrifications or currents in these bodies”—which was fine if you assumed that electric and magnetic forces acted at a distance—partial differential equations show how things change through the whole space around the bodies. So, he suggested, such equations are the natural tools for expressing the way changing electric and magnetic forces and fluxes produce electromagnetic effects. The field, whatever it was made of, mediated these changes through space—just as the field of wheat mediates a breeze rippling through it like a wave.19

• • •

• • •

麦克斯韦是第一个将法拉第发现的磁感应(即相对运动的磁铁感应出电流)转化为符号数学形式的人。将磁铁穿过线圈或将线圈穿过磁场会改变磁通量(或者法拉第认为是从磁铁穿过线圈的“力线”数量,如图6.2所示);麦克斯韦必须找到正确的方程来将这种变化的磁通量与感应电流量联系起来。这今天被称为“法拉第定律”。

Maxwell was the first to put Faraday’s discovery of magnetic induction— the inducing of electric current by a relatively moving magnet—into symbolic mathematical form. Moving the magnet through a coil of wire, or turning the coil through a magnetic field, changes the magnetic flux—or what Faraday thought of as the number of “lines of force” from the magnet through a loop of wire, as you can see in figure 6.2; Maxwell had to find the right equations for relating this changing magnetic flux to the amount of current induced. This is known today as “Faraday’s law.”

然后是“安培定律”,这是奥斯特发现的数学版本,用变化的电流产生的磁力来表示。麦克斯韦通过更仔细地定义电流,然后展示它与变化的电通量的关系,扩展了安培在这方面的工作。20之前谈到通量是表面积分,安培也使用过表面(和线)积分,因为他一直从事超距作用传统。因此,麦克斯韦不仅需要完成安培的结果,还需要将这些积分重写为导数,以表达场的概念。

Then there was “Ampère’s law,” the mathematical version of Øersted’s discovery, in terms of the magnetic force produced by a changing electric current. Maxwell extended Ampère’s work on this by defining current more carefully, and then showing how this relates to changing electric flux.20 I spoke before about flux as a surface integral, and Ampère had used surface (and line) integrals, too, for he had worked in the action-at-a-distance tradition. So, Maxwell needed not only to complete Ampère’s results but also to rewrite these integrals as derivatives, in order to express the field concept.

还有许多内容需要描述和定义,包括静电和磁力的高斯-库仑定律,麦克斯韦现在必须用场的语言重写它们。即使电荷或磁铁静止不动,平方反比定律也会显示它们各自发出的力如何随距离而变化(“静态”表示力不会随时间而变化)——因此,导数也是这些定律的自然语言选择。在条形磁铁周围的磁场中,你可以看到铁屑排列的变化,铁屑聚集在力最强的两极附近,但随着距离的增加而减弱,铁屑向外扩散——如图6.3a6.3c所示。

There was much more to describe and define—including the Gauss-Coulomb laws for static electricity and magnetism, which Maxwell now had to rewrite in field language. For even when you have a charge or a magnet just sitting still, the inverse square law shows how the force emanating from each of them changes with distance (“static” means the force doesn’t change in time)—so derivatives are a natural choice of language for these laws, too. In the magnetic field around a bar magnet, you can see these changes in the alignment of iron filings, the filings bunching up near the poles where the force is strongest but spreading out as it weakens with distance—as in figures 6.3a and 6.3c.

麦克斯韦从传统的超距作用积分转变为偏微分场方程的核心在于“积分学基本定理”,你可能还记得学校里学过这个定理,不过我会在下一个尾注中详细说明。你也会从中了解到麦克斯韦对斯托克斯定理的巧妙运用,该定理曾出现在史密斯奖考试中——你会看到他是如何运用现在的后矢量的,被称为“散度定理”,由高斯、格林和俄罗斯数学家米哈伊尔·奥斯特拉格拉德斯基首创。21

The nub of how Maxwell changed from traditional action-at-a-distance integrals to partial differential field equations lies in the “fundamental theorem of integral calculus,” which you may remember from school, although I’ll spell it out in the next endnote. You’ll also get a hint there of the marvelous use Maxwell made of Stokes’s theorem, which had featured in the Smith’s Prize exam—and you’ll see how he used what is now, post-vectors, known as the “divergence theorem,” pioneered by Gauss, Green, and the Russian mathematician Mikhail Ostragradsky.21

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图 6.3C 。事后看来,你可以看到图 6.3a中所示的物理磁屑线如何以矢量场的形式进行数学表示。箭头指向该点的力的方向,其长度表示力的强度——因此它们在靠近磁极的地方较长,因为那里的力最强。

FIGURE 6.3C. In hindsight, you can see how the physical lines of filings shown in figure 6.3a can be mathematically represented as a vector field. The arrows point in the direction of the force at that point, and their length indicates the strength of the force—so they are longer close to the magnet’s poles, where the force is strongest.

我们将在下一章中看到麦克斯韦方程组,但这里的重点是,首先,在他关于电磁场的里程碑式论文中——他在 1864 年底提交给皇家学会,并于 1865 年 1 月在其《哲学学报》上发表——他首次正式统一了电和磁。

We’ll see Maxwell’s equations in the next chapter, but the point here is that first, in his landmark paper on the electromagnetic field—which he presented to the Royal Society at the end of 1864 and published in its Philosophical Transactions in January 1865—he formally united electricity and magnetism for the first time.

其次,与高斯、斯托克斯等人一样,麦克斯韦只使用了斯托克斯定理和散度定理的分量形式(他没有以当今教科书中的紧凑全矢量形式来陈述它们)。因此,在他 1865 年的论文中,他发现了他为描述电磁场而定义的各种量的分量之间的关系换句话说,他还没有以完整的矢量形式阐述他的理论。不过,他确实谈到了既有方向又有大小的量,所以他创建了一个矢量电磁理论。在泰特的帮助下,他只用了几年时间就得到了汉密尔顿全矢量微积分。

Second, like Gauss, Stokes, et al., Maxwell used only the component form of Stokes’s theorem and the divergence theorem (he didn’t state them in the compact whole-vector form you see in textbooks today). So, in his 1865 paper, he found relationships between the components of the various quantities he’d defined to describe the electromagnetic field. In other words, he had not yet formulated his theory in full vector form. He did talk about quantities having both direction and magnitude, though, so he had created a vectorial electromagnetic theory. With Tait’s help, he would get to Hamiltonian whole-vector calculus in just a few years’ time.

1865 年是美国历史上以及物理学和数学史上具有重要意义的一年,因为血腥的内战终于结束了。帮助转化麦克斯韦方程的人之一最早将现代(后汉密尔顿)向量符号引入到现代向量符号的人是约西亚·威拉德·吉布斯,他的父亲长期以来一直致力于废除奴隶制。战争期间,年轻的吉布斯在耶鲁大学攻读工程学博士学位——这是美国第一个工程学博士学位。又过了十年,他才发现麦克斯韦,又过了几乎十年,他才在向量史上留下自己的印记。所以,我在这个故事中试图展示的部分内容是,思想的发展和找到最佳形式需要多长时间。我认为,了解这段漫长的旅程可以帮助那些难以掌握一个想法的学生,因为他们会发现早期的数学家也曾经历过挣扎。但它也可以帮助展示我们现在可能认为理所当然的具体思想的力量——而在认为它们理所当然的过程中,我们可能会错过其中的一些力量。

The year 1865 is a momentous one in American history as well as in physics and maths history, for the bloody Civil War finally ended. It is fitting that one of the men who would help transform Maxwell’s equations into modern (post-Hamilton) vector notation is Josiah Willard Gibbs, whose father had long agitated for the abolition of slavery. During the war, young Gibbs was completing his engineering doctorate at Yale—the very first American engineering doctorate. It would be another decade before he discovered Maxwell, and almost another still before he made his mark in the history of vectors. So, part of what I’ve been trying to show throughout this story is just how long it takes for ideas to develop and find their best form. I think that knowing something of this long journey can help students who are struggling to grasp an idea, by showing that earlier mathematicians had struggled, too. But it can also help show the power of specific ideas that we might now take for granted—and in taking them for granted, we might miss some of that power.

例如,麦克斯韦选择用微分方程来表示电磁行为,但这些方程在数学上等同于早期电磁先驱们如果成功的话可能开发的积分方程——事实上,麦克斯韦方程的积分形式今天经常使用。(这种等价性正是上一个尾注中的“基本定理”的用武之地。)区别在于解释,尽管今天,无论你使用哪种形式的微积分,的概念在物理学中都是至关重要的——正如法拉第、汤姆森和麦克斯韦所表明的那样,“场”只是特定量在空间中的值分布。例如,你周围空气温度的记录是一个“标量场”——一组表示空间中每个点温度大小的数字——而作用于整个场中小单位电荷或铁屑的电力和磁力是“矢量场”,因为在每个点,这些力都有大小和方向。 (换句话说,矢量和矢量场之间的区别在于,矢量在给定点处定义,而矢量场为相关空间中的每个点分配一个矢量。)但麦克斯韦选择了微积分,这不仅使他能够从数学上定义电磁场,而且还做出了令人惊讶且影响深远的发现——这项发现彻底改变了我们彼此以及与宇宙本身的交流方式。

For instance, Maxwell chose to represent the behavior of electromagnetism using differential equations, but these are mathematically equivalent to the integral equations that the early electromagnetic pioneers might have developed had they succeeded in their quest—in fact, the integral form of Maxwell’s equations is often used today. (This equivalence is where the “fundamental theorem” in the previous endnote comes into it.) The difference is in the interpretation, although today, the idea of a field is paramount in physics, no matter which form of calculus you use—for as Faraday, Thomson, and Maxwell showed, a “field” is just the distribution of values of a particular quantity through space. For example, a record of the temperature of the air around you right now is a “scalar field”—a set of numbers representing the magnitude of the temperature at each point in space—while the electric and magnetic forces acting on little unit test charges or filings placed throughout the field are “vector fields,” because at each point these forces have both magnitude and direction. (In other words, the difference between a vector and a vector field is that a vector is defined at a given point, and a vector field assigns a vector to each point in the relevant space.) But it was Maxwell’s choice of differential calculus that enabled him not only to define electromagnetic fields mathematically, but also to make a surprising and far-reaching discovery—one that has revolutionised the way we communicate with each other, and with the universe itself.

不可思议的推论

AN UNCANNY DEDUCTION

经过十年的艰苦工作和深思熟虑,麦克斯韦于 1865 年发表了电磁场方程,正式将电和磁统一起来。尽管这是一项伟大的成就,但这并不是终点,因为他还从场方程中推导出电磁效应不会远处发生的原因。它们以横波的形式传播,与托马斯·杨通过著名的双缝实验为光确定的波相同。

When Maxwell published his electromagnetic field equations in 1865, after ten years of hard work and deep thinking, he had formally united electricity and magnetism. Yet magnificent achievement that it was, this was not the end of it, for he also deduced from his field equations the reason that electromagnetic effects do not occur at a distance. They are propagated as transverse waves, the same kind of wave that Thomas Young had identified for light via that famous two-slit experiment.

更重要的是,麦克斯韦发现他理论中的电磁波也几乎与光速相同。(我说“几乎”是因为当时很难进行精确测量。)这是一个不容忽视的巧合——它证明了麦克斯韦对导数的直觉既深刻又简单。最初的横向“波动方程”是一个微分方程,一个世纪前就被推导出来描述拨弦的振动方式——但这个引人注目的新数学平行表明,正如麦克斯韦谦虚地说的那样,“光本身(包括辐射热,以及其他辐射,如果有的话)是一种电磁扰动,以波的形式通过电磁场传播,遵循电磁定律。”他一举将电和磁统一起来,也把光统一起来了。他甚至指出了“其他辐射”的可能存在,这一诱人的预测让海因里希·赫兹等人走上了发现无线电波的道路。

What’s more, Maxwell found his theoretical electromagnetic waves had just about the same speed as light, too. (I say “just about” because of the difficulty in making accurate measurements at the time.) It was a coincidence too delicious to ignore—and it proved that Maxwell’s hunch about derivatives was as profound as it was simple. The original transverse “wave equation,” a differential equation, had been derived a century earlier to describe the way a plucked string vibrates—but this remarkable new mathematical parallel suggested, as Maxwell put it modestly, “that light itself (including radiant heat, and other radiations if any) is an electromagnetic disturbance in the form of waves propagated through the electromagnetic field according to electromagnetic laws.” In one fell swoop he had united not just electricity and magnetism but light, too. He’d even pointed to the possible existence of “other radiations,” a tantalising prediction that set the likes of Heinrich Hertz on the path to discovering radio waves.

正如爱因斯坦后来写道,想象一下麦克斯韦在电磁学和光之间建立联系时的感受!“世界上很少有人能有这样的经历,”他补充道。麦克斯韦本人在 1865 年 1 月写给他表弟的一封信中,以他一贯低调的方式私下表达了他的兴奋之情:他几乎是顺便补充说:“我还有一篇论文,里面有关于光的电磁理论,除非我确信相反,否则我认为它是伟大的武器。”他不喜欢吹嘘,所以你可以感觉到当他做出这个不可思议的推论时,他一定是多么激动。22

As Einstein later wrote, imagine Maxwell’s feelings when he made the connection between electromagnetism and light! “To few men in the world has such an experience been vouchsafed,” he added. Maxwell himself had privately expressed his excitement in his typically understated way, in a letter to his cousin in January 1865: almost as an aside he’d added, “I have also a paper afloat, with an electromagnetic theory of light, which, till I am convinced to the contrary, I hold to be great guns.” He was not given to boasting, so you can sense how thrilling it must have been when he made his uncanny deduction.22

麦克斯韦之所以能发现光的电磁性质,是因为他巧妙地选择了数学语言。23起初,人们并没有热烈地接受它。主流面临的问题是,法拉第的场的概念已经表达出来只有细致的数据,没有方程,现在麦克斯韦的理论完全用方程来表达——没有物理模型来理解场到底什么。这种反应也曾困扰着牛顿的引力理论。在早期的多篇论文中,麦克斯韦确实提出了几种力学模型来解释场是什么以及场是如何传递电磁效应的,但他知道没有办法证明其中任何一个的存在。他相信,场的后果的数学知识应该足够了——就像牛顿知道,尽管他没有解释引力本身,但只要他的方程准确地描述引力的效应就足够了。

Maxwell’s momentous discovery of the electromagnetic nature of light was only possible because of his clever choice of mathematical language.23 Not that it met with rapturous acceptance at first. The problem for the mainstream was that Faraday’s conception of the field had been expressed with meticulous data but no equations, and now Maxwell’s theory was expressed entirely in terms of equations—with no physical model for conceiving what the field actually was. It was the very same response that had dogged Newton’s theory of gravity. In various earlier papers, Maxwell had indeed suggested several mechanical models that might explain what the field was and how it transmitted electromagnetic effects, but he knew there was no way of proving the existence of any of them. The mathematics of the field’s consequences, he believed, should be enough—just as Newton had known it was enough that his equations accurately described the effects of gravity, even though he hadn’t explained gravity itself.

尽管如此,麦克斯韦仍在思考如何更清楚地表达他的场。在 1865 年的论文开头,他说找到电磁理论的第一步是“确定物体之间作用力的强度和方向”——就像牛顿最初开始定义力时所做的那样。这是一种矢量方法,但还不是完整的矢量理论,因为麦克斯韦还没有研究四元数。但这种情况即将改变——在他的老朋友泰特的帮助下。

Still, Maxwell kept on thinking about how he could express his fields more clearly. At the beginning of his 1865 paper, he’d said that the first step in finding a theory of electromagnetism had been to “ascertain the strength and direction of the forces acting between the bodies”—just as Newton had done when setting out to define force in the first place. It was a vectorial approach, but not yet a full vector theory, for Maxwell hadn’t yet studied quaternions. But that was about to change—with help from his old friend Tait.

四元数终于来了

QUATERNIONS AT LAST

当麦克斯韦致力于发展法拉第场的思想时,泰特则在研究热的数学。如果T是温度,那么“热方程”将T的拉普拉斯算子与电视,因此它显示了温度如何随时间变化而随空间变化——例如,当你从厨房端起早晨的咖啡时,它会随着温度的冷却而变化。傅立叶于 1822 年发表了热方程,然后在气候科学的奠基文献——他 1827 年的论文《论地球和行星际空间的温度》中使用了它。正如我所提到的,汤姆森通过改编傅立叶关于热的工作,首次提出了电场类比,今天,它的公式显示了物质如何扩散,有无数的用途,从工程到政策制定(因为思想也可以扩散)。无论如何,1857 年,泰特突然想起了四年前那次狩猎旅行中他在汉密尔顿的《四元数讲座》中读到的东西。汉密尔顿将拉普拉斯算子转化为矢量形式,通过定义一个“矢量算子”,他将其表示为就像这样(只不过他仍然使用普通的 d 而不是当时才刚刚开始流行的现代花括号):

While Maxwell had been developing Faraday’s field idea, Tait had been studying the mathematics of heat. If T is the temperature, the “heat equation” relates the Laplacian of T to Tt, so it shows how the temperature changes through space as it changes in time—through your morning coffee, for example, as it cools while you carry it from the kitchen. Fourier had published the heat equation in 1822, and then used it in the founding document of climate science, his 1827 paper, “On the Temperatures of the Terrestrial Sphere and Interplanetary Space.” Thomson had made the first electric field analogy by adapting Fourier’s work on heat, as I mentioned, and today its formula showing how things diffuse has myriad uses, from engineering to policymaking (for ideas can diffuse, too). Anyway, in 1857, Tait suddenly remembered something he’d read in Hamilton’s Lectures on Quaternions on that hunting trip four years earlier. Hamilton had cast the Laplacian into vector form, by defining a “vector operator” that he expressed like this (except he still used ordinary d’s instead of the modern curly ones that were only beginning to gain ground):

=++

=ix+jy+kz.

如果你学过矢量微积分,你就会知道这是

If you’ve studied vector calculus, you’ll recognise this as

=++

=xi+yj+zk

用今天的符号表示。(正如我在序言中提到的,还有其他现代的方式来表示向量,比如在字母顶部加帽子或横线,或者在下面加波浪线,但粗体既醒目又容易输入,所以在现代教科书和论文中很常见。)无论哪种方式,你处理的都是笛卡尔坐标系,所以我们很快就会看到,无论你使用单位虚数i、j、k还是单位向量i、j、k作为向量运算符的基础,你基本上都会得到相同的结果。

in today’s notation. (As I mentioned in the prologue, there are other modern ways of denoting vectors, such as hats or bars on top of a letter or squiggles underneath, but boldface is both arresting and easy to type, so it is common in modern textbooks and papers.) Either way, though, you’re dealing with a Cartesian coordinate system, so we’ll see shortly that you essentially get the same results, no matter whether you use the unit imaginary numbers i, j, k or the unit vectors i, j, k as the basis of your vector operator.

泰特立即着手将这种新型算符应用到他的热和电研究中,并将汉密尔顿的符号 ⊲ 变成了 ∇。泰特还为这个符号取了一个名字“nabla”,这是他的助手威廉·罗伯逊·史密斯建议的,因为“nabla”是希腊语中古亚述竖琴的拉丁音译,其形状与 ∇ 类似,呈倒三角形。(后来,吉布斯将汉密尔顿的“nabla”替换为“del”——显然是因为 ∇ 也类似于希腊字母“delta”的倒置版本。这两个名字今天仍在使用。)至于现代表示 ∇ 的粗体字,当反传统的奥利弗·赫维赛德最终进入我们的故事时,它将载入史册。但我在这里介绍它,是因为我认为它可以让现代读者更容易地看到,正如泰特所做的那样,汉密尔顿的算符本身是矢量的。

Tait immediately set to work applying this new kind of operator in his work on heat and electricity, turning Hamilton’s symbol ⊲ into ∇ in the process. Tait also linked this symbol with a name, “nabla,” which his assistant William Robertson Smith suggested, because “nabla” is the Latin transliteration of the Greek word for an ancient Assyrian harp, which had an inverted triangle shape similar to ∇. (Later, Gibbs will replace Hamilton’s “nabla” with “del”—apparently because ∇ also resembles an upside-down version of the Greek letter “delta.” Both names are used today.) As for the boldface type in the modern representation of ∇, it will make its way into history when the iconoclastic Oliver Heaviside eventually enters our story. But I’m introducing it here because I think it makes it easier for modern readers to see, as Tait certainly did, that Hamilton’s operator itself is vectorial.

例如,这意味着它可以将普通(标量)函数(例如势)转换为矢量。前面我提到,如果力具有势V,则力的笛卡尔分量为。这些只是您插入后获得的组件=++ 变成现代矢量版本,=++。因此,我们不必通过列出力的组成部分来写力,

This means, for example, that it can turn an ordinary (scalar) function, such as the potential, into a vector. Earlier I mentioned that if a force has a potential V, the Cartesian components of the force are Vx,vy,Vz. These are just the components you get if you insert =ix+jy+kz or into the modern vector version, =xi+yj+zk. So instead of writing the force by listing its components,

F=

F=Vx,Vy,Vz,

你可以更简单、更透明地写成

you can write it much more simply and transparently as

F = ∇ V

F = ∇V,

F = ∇ V,正如 Tait 和后来的麦克斯韦所写。

or F = ∇V as Tait, and later Maxwell, wrote it.

它更简单,因为它是一种更紧凑的表示F的方式,在编写或编程时节省时间和空间。它更透明,因为它将力表示为一个整体矢量,而不是单独列出其分量。这使得更容易看出问题是关于力和势能的物理概念,而不仅仅是数字(分量)值的列表。这是一个微妙的点,我们故事中的一些玩家需要很长时间才能理解它。

It’s simpler because it’s a more compact way of representing F, saving time and space when you are writing it out or programming it. And it’s more transparent because it expresses the force as a whole vector, rather than listing its components separately. This makes it easier to see that the problem is about force and potential as physical concepts, rather than just a list of numerical (component) values. This is a subtle point, and it will take some of the players in our story a long time to understand it.

抛开物理学不谈,由于 ∇ 是矢量,你也可以将其与自身形成标量积,将拉普拉斯算子写为 ∇ ∙ ∇,或者哈密顿所写的 ∇ 2,我们今天也这样做。因此,例如,拉普拉斯方程变成

Physics aside, since ∇ is vectorial, you can also form its scalar product with itself, writing the Laplacian as ∇ ∙ ∇, or ∇2 as Hamilton wrote it, and as we also do today. So, for example, Laplace’s equation becomes

∇2V = 0

2V = 0,

这比22+22+22=0

which is neater than 2Vx2+2Vy2+2Vz2=0.

你也可以取 nabla 和另一个向量的标量积——也可以取向量或叉积。下一章中我们将看到更多这方面的内容。这将使 Tait 能够使用这些 nabla 运算以全向量微积分的形式写出斯托克斯定理和散度定理——他将是第一个这样做的人。不过,他首先全身心地投入到汉密尔顿的书中。他对四元数及其向量微积分算子在物理学中的作用感到非常兴奋,因此他问汉密尔顿是否介意与他通信。汉密尔顿当然很高兴找到了一个数学上的志趣相投的人。在 1858 年他们开始通信时,他告诉 Tait那是一个奇妙的日子,四元数的秘密在布鲁姆桥上一闪而过,“我的孩子们从此就把这座桥叫做四元数桥。” 24

You can also take the scalar product of nabla and another vector—and you can take the vector or cross product, too. We’ll see more of this in the next chapter. It will enable Tait to use these nabla operations to write Stokes’s theorem and the divergence theorem in terms of whole-vector calculus—and he’ll be the first to do this. First, though, he threw himself into Hamilton’s book. He was so excited about the role that quaternions and their vector calculus operators could play in physics that he asked if Hamilton would mind corresponding with him. Hamilton, of course, was delighted to find a mathematical kindred soul. At the beginning of their correspondence in 1858, he told Tait of that marvelous day when the secret of quaternions revealed itself in a flash at Broome Bridge, “which,” he added, “my boys have since called the Quaternion Bridge.”24

到了 1859 年,汉密尔顿已经可以自在地与泰特分享一些私人感受,就像他与德·摩根一样——包括讲述他的初恋凯瑟琳·迪斯尼嫁给别人时他所感到的绝望。(多年后,她告诉汉密尔顿,她只爱他一个人,她的父母强迫她接受一段更“合适”但最终不幸福的婚姻。)但正如他告诉泰特的那样,“那些日子已经过去了:幸福吗?是的,就变得更理智、更少感性而言。” 25看来他读过简·奥斯汀的《理智与情感》!无论如何,54 岁的汉密尔顿很满足。

By 1859 Hamilton felt comfortable enough to share quite personal feelings with Tait, as he did with De Morgan—including recounting the despair he’d felt when his first love Catherine Disney had married someone else. (It was years later that she told Hamilton she had loved only him, and that her parents had pressured her into a more “suitable,” ultimately unhappy marriage.) But as he told Tait, “Those days are over: happily? Yes, so far as getting a little more sense, and less sensibility, is concerned.”25 It seems he’d read Jane Austen’s Sense and Sensibility! At any rate, at the age of fifty-four, Hamilton was content.

28 岁的泰特似乎也很满足。两年前,他与玛格丽特·波特结婚,也陶醉于工作。除了在贝尔法斯特教书之外,他还在 1859 年发表了第一篇关于四元数的论文(论波)。然后,在 1860 年,他晋升为爱丁堡大学的自然哲学教授,当时他和麦克斯韦都就读于这所大学。麦克斯韦也申请过这个职位,但不管对错,人们都认为泰特更胜一筹。JM 巴里在成为《彼得潘》的作者之前是泰特的学生之一,他声称泰特是有史以来最出色的演示者。他回忆说,上课时,泰特“闪烁的小眼睛里闪烁着迷人的光芒……我曾看到过一个人在泰特的目光下惊慌失措。”然而,巴里继续说,那双眼睛也可以“像男孩一样快乐”——尤其是当“他把一管水倒在一群坚持挤在实验太近的学生身上时”。但麦克斯韦也有他的学生粉丝——例如,未来的天文学家大卫·吉尔描述了麦克斯韦在讲座结束后会和感兴趣的学生呆上几个小时。至于泰特和麦克斯韦本人,从他们之间的信件来看,这种职业竞争并没有影响他们的友谊。毕竟,他们从学生时代起就是朋友和对手。26

Twenty-eight-year-old Tait seemed content, too. He had married Margaret Porter two years earlier, and he was also reveling in his work. As well as his teaching in Belfast, he published his first quaternion paper (on waves) in 1859. Then, in 1860, he won a promotion, to professor of natural philosophy at his and Maxwell’s old university, Edinburgh. Maxwell had applied for the post, too, but rightly or wrongly Tait was seen as the better lecturer. J. M. Barrie was one of Tait’s students, before he became famous as the author of Peter Pan, and he claimed that Tait was the most superb demonstrator ever. He recalled that during classes Tait’s “small twinkling eyes had a fascinating gleam in them…. I have seen a man fall back in alarm under Tait’s eyes.” Yet, Barrie continued, those eyes could also “be merry as a boy’s”—especially when “he turned a tube of water on a crowd of students who would insist on crowding too near an experiment.” But Maxwell had his student fans, too—for instance, the future astronomer David Gill described how Maxwell would stay behind with interested students for hours after the lecture. As for Tait and Maxwell themselves, such professional rivalry had no effect on their friendship, judging by their letters to each other. After all, they’d been friends and rivals since school.26

我们将在下一章中看到他们之间古怪的通信,因为 Tait 即将因其在四元数方面的工作而出名,而 Maxwell 即将向他提出大量问题。他将在他的《电磁论》中使用答案——这本非凡的书将为现代矢量微积分之路的最后一步奠定基础。

We’ll see something of their quirky correspondence in the next chapter, for Tait is about to become famous for his work on quaternions, and Maxwell is about to pepper him with questions. He will use the answers in his Treatise on Electricity and Magnetism—and this extraordinary book will set the scene for the final step on the road to modern vector calculus.

(7)从四元数到向量的缓慢历程

(7) THE SLOW JOURNEY FROM QUATERNIONS TO VECTORS

泰特着迷于四元数(更确切地说,四元数的矢量部分)如何使他能够更紧凑地写出物理方程,尤其是当他应用哈密顿的算符 ∇(“nabla”)时。其他人(从牛顿到麦克斯韦)一直在为力或速度等矢量的每个分量写出单独的方程。哈密顿本人专注于数学而不是物理,因此他很高兴看到泰特为他的四元数找到了新用途——泰特从贝尔法斯特来到都柏林,以便与哈密顿亲自交谈。他们还交换了数十封信,包括哈密顿的欣喜信,希望有一天他和泰特会因他们的努力而得到感谢。汉密尔顿确实得到了认可,但没有真正感谢他在这个故事中所扮演的角色——但当他于 1865 年去世时,就在麦克斯韦发表开创性的电磁场理论几个月后,34 岁的泰特准备接替他继续工作。

Tait was fascinated by the way quaternions—or rather, the vector part of a quaternion—enabled him to write physics equations so much more compactly, especially when he applied Hamilton’s operator ∇ (“nabla”). Everyone else—from Newton to Maxwell—had been writing out separate equations for each component of a vectorial quantity such as force or velocity. Hamilton himself had focussed on maths rather than physics, so he was delighted by the new uses Tait was finding for his quaternions—and Tait had traveled from Belfast down to Dublin so that he and Hamilton could talk in person. They also exchanged dozens of letters, including Hamilton’s joyful letter hoping that someday he and Tait would be thanked for their efforts. Hamilton did receive recognition but no real thanks for his part in the story—but when he died in 1865, just a few months after Maxwell published his trailblazing electromagnetic field theory, thirtyfour-year-old Tait was ready to carry on in his stead.

首先,他为他的朋友和导师写了一份长达 37 页的讣告——几个月后,也就是 1866 年,它出现在《北不列颠评论》上。他开敦促读者在学术生活中主宰的“争论的喧嚣”和“知识巨人的争斗”中停下来,“在沉思某种更高尚、更微妙的东西中寻求安宁:一个天才的性格和作品 ”是的,他继续说,“天才”这个词已经被滥用了,因为它指的是比聪明才智更有创造力和独创性的东西——但汉密尔顿一位最高级别的天才,值得与“所有时代和国家的最伟大天才,如拉格朗日和牛顿”相提并论。因为汉密尔顿的贡献远不止四元数分析,尽管它很了不起。泰特说,他早期关于光学和动力学的论文显示出“对符号的掌握和数学语言的流畅(如果可以使用这种表达方式的话)几乎无人能及。”

First up, he wrote a 37-page obituary for his friend and mentor—it appeared in the North British Review a few months later, in 1866. He opened by urging his readers to pause amid the “din of controversy” and “the battle of intellectual giants” that dominated academic life, and “seek repose in the contemplation of something far more elevated and much more subtle: the character and works of a man of genius.” Yes, he continued, the word “genius” has become overused, because it refers to something far more creative and original than mere cleverness—but Hamilton was a genius of the highest order, fit to rank alongside “those of the grandest of all ages and countries, such as Lagrange and Newton.” For Hamilton contributed far more than quaternion analysis, marvelous though it was. His early papers on optics and dynamics, Tait said, showed “a mastery over symbols, and a flow of mathematical language (if the expression can be used) almost unequalled.”

1867 年,泰特出版了自己的《四元数初等论》,这是第一本(相对)易懂的矢量方法教科书。事实上,除了符号(他效仿汉密尔顿使用希腊字母表示矢量,用SV代替现代的点和叉表示标量和矢量积)之外,这本书读起来与现代矢量分析教材没什么不同。然而,随着汉密尔顿的离去和新合作者的出现,他应用四元数的热情……并未减弱,因为他在接下来的五年里发表了更多关于这个主题的研究论文,只是有所减弱。因为他的新合作者讨厌四元数!

In 1867, Tait published his own Elementary Treatise on Quaternions, the first (relatively) accessible textbook on vectorial methods. In fact, aside from the notation—he follows Hamilton in using Greek letters for vectors, and S and V instead of the modern dot and cross for scalar and vector products—it reads not unlike a modern vector analysis text. Yet with Hamilton gone and a new collaborator on the scene, his passion for applying quaternions was … not dampened, because he published several more research papers on the subject over the next five years, but somewhat curtailed. For his new collaborator hated quaternions!

我指的是威廉·汤姆森——麦克斯韦的早期导师,未来的开尔文勋爵。他和泰特在 1861 年相识,一年后泰特从贝尔法斯特搬回家乡爱丁堡大学。汤姆森出生于贝尔法斯特,但在格拉斯哥长大,年仅 22 岁时就成为格拉斯哥大学的教授。然而,尽管他发表了一些杰出的数学论文——包括 17 岁时写的关于法拉第场的论文——但汤姆森很快就凭借技术成就而非数学成就而变得富有和出名。例如,在 19 世纪 50 年代末,他曾担任第一条大西洋海底电报电缆铺设工程的首席技术顾问——第一条电缆断裂时他在船上,另一次是一场可怕的风暴,这场风暴威胁着整个项目。最终,电缆铺设完毕,身为北大西洋电报公司董事的汤姆森从欧洲向美国发送了第一批电报信号。他还发明了许多有利可图的电工仪器,并帮助开创了热力学科学——开尔文温标就是以他的名字命名的。在如此多的事情同时进行的情况下,他竟然还能抽出时间与泰特合作,这真是令人惊讶:他们正在编写一本新的物理教科书《自然哲学论》。“自然哲学”仍是理论物理学的名称,他们两人都是这门学科的教授。

I’m speaking of William Thomson—Maxwell’s early mentor, and the future Lord Kelvin. He and Tait had met in 1861, a year after Tait moved from Belfast back home to the University of Edinburgh. Thomson was Belfast-born but Glasgow-raised, and he’d become a professor at the University of Glasgow when he was only twenty-two. Yet despite some remarkable mathematical papers—including the one on Faraday’s field idea, written when he was just seventeen—Thomson would soon become rich and famous for his technical rather than his mathematical achievements. In the late 1850s, for example, he’d been the chief technical advisor during the dramatic laying of the very first telegraph cable under the Atlantic— he’d been on board ship when the first cable broke, and again during a terrible storm that threatened to derail the whole project. Eventually the cable was laid, and Thomson, who was a director of the North Atlantic Telegraph Company, sent the first telegraphic signals from Europe to America. He also invented many profitable electrical instruments and helped pioneer the science of thermodynamics—the Kelvin temperature scale is named after him. With so many irons in the fire, it’s amazing he still managed to find time to collaborate with Tait: they were writing a new physics textbook, A Treatise on Natural Philosophy. “Natural philosophy” was the name still being used for theoretical physics, and it was the subject in which they were both professors.

然而,从 Tait 的角度来看,Thomson 实际上投入到他们的项目上的时间相当少。“我对这本伟大的书感到厌倦,”Tait 在 1864 年写信给他。“如果你只是偶尔给我寄一些零碎的东西……我该怎么办?你甚至没有给我一点提示,说明你想在我们目前关于液体和气体静力学的章节中做什么!”Thomson 当时在德国,Tait 接着说,虽然他几乎完成了关于粒子动力学的一章,但他当然不想支付 45 倍的邮费将其寄到海外,因为 Thomson 很可能要等到回到苏格兰才能读到它。1

From Tait’s point of view, however, the amount of time Thomson actually devoted to their project was rather small. “I am getting sick of the great Book,” Tait had written to him in 1864. “If you send me only scraps … at rare intervals, what can I do? You have not given me even a hint as to what you want done in our present chapter about statics of liquids and gases!” Thomson was then away in Germany, and Tait went on to say that although he’d almost finished a chapter on the kinetics of particles, he certainly didn’t feel like paying forty-five times more postage to send it overseas when in all probability Thomson wouldn’t get around to reading it till he was back in Scotland.1

泰特当时还在为他同时撰写的《四元数初等论》写一章关于运动学的内容。他告诉读者,虽然四元数微积分可能很难学,但它值得掌握,因为它“为高级学生提供了最特别的优势,不仅可以帮助他们解决复杂的问题,而且还能提供宝贵的智力训练”。但汤姆森对此不以为然。当然,他使用了矢量分量方程,但他认为根本不需要汉密尔顿的纳布拉算子允许的全矢量微积分。他指出,无论如何你都会用分量进行计算——例如,在标量积中,当你计算一个b=一个1b1+一个2b2+一个3b3或者当你通过计算证明 ∇ 2 V = 022+22+22或者在物理和工程问题中,当你将一个向量“分解”为其分量时(如图 8.1所示)——仅举两个例子。因此,对于他们的合作项目,Tait 不情愿地推迟了:四元数被淘汰了。

Tait had also been working on a chapter on kinematics for his Elementary Treatise on Quaternions, which he’d been writing at the same time. He told his readers that although quaternion calculus may be difficult to learn, it is worth mastering because it offers “the most extraordinary advantage to the advanced student, not alone as aiding him in the solution of complex questions, but as affording an invaluable mental discipline.” But Thomson was having none of it. He used vectorial component equations, of course, but he saw no need whatsoever for the whole-vector calculus that Hamilton’s nabla operator allowed. He made the good point that you carried out computations with components anyway—in the scalar product, for example, when you calculate ab=a1b1+a2b2+a3b3. Or when you show that ∇2V = 0 by calculating 2Vx2+2Vy2+2Vz2, or in physics and engineering problems when you “resolve” a vector into its components (as in fig. 8.1)—to take just two more examples. So, for their joint project, Tait reluctantly deferred: quaternions were banished.

尽管两人意见不一,但汤姆森注意到,泰特总是愿意用莎士比亚、狄更斯和萨克雷的“精彩语录”来“点亮”他们紧张的数学合作。有时,两人会在泰特的小书房里一起工作,煤气灯的灯光在已经被烟雾熏黑的墙上投下怪异的阴影。那些昏暗的墙上还贴着泰特用木炭写下的当时最伟大的科学家名单:汉密尔顿排在第一位,法拉第紧随其后(法拉第于 1867 年去世)。泰特和汤姆森曾讨论过他们的朋友麦克斯韦——用汤姆森的话说,他是“一颗冉冉升起的巨星”——但他们认为他还太年轻,无法进入这份神圣的名单。2

Despite their disagreements, Thomson noted that Tait was always ready to “brighten” their intense collaborative mathematical labour with “delightful quotations” from Shakespeare, Dickens, and Thackeray. Sometimes the two men worked together in Tait’s small home study, by the light of a gas lamp that cast eerie shadows on a wall already dimmed by tobacco smoke. Those dingy walls also contained a list Tait had written in charcoal of the greatest scientists then living: Hamilton was first, followed by Faraday (who died in 1867). Tait and Thomson had discussed their friend Maxwell—“a rising star of the first magnitude,” as Thomson put it—but thought him still too young to make the hallowed list.2

最后,汤姆森和泰特的《自然哲学论文》终于可以出版了。这本书于 1867 年问世,由于其前沿内容(包括新兴的热力学)而具有巨大影响力。它甚至还讨论了用于复杂数学计算的计算机器。查尔斯·巴贝奇和艾达·洛夫莱斯在 19 世纪 40 年代就提出了这种机器的真正可能性,但巴贝奇的“机器”从未真正投入使用,因此汤姆森描述了他自己的理论设计。这本书被俗称为TT′(发音为“ TT -dash”),汤姆森(T)和泰特(T′)的昵称也沿用至今。

At last, Thomson and Tait’s Treatise on Natural Philosophy was ready for the publisher. It saw the light of day in 1867 and would prove very influential because of its cutting-edge content, including the emerging science of thermodynamics. It even included a discussion on calculating machines for sophisticated mathematical computations. Charles Babbage and Ada Lovelace had brought the real possibility of such a machine to attention in the 1840s, but Babbage’s “engine” never quite got off the ground, so Thomson described his own theoretical designs. The book was colloquially known as T and T′—pronounced “T and T-dash”—and the nicknames stuck for Thomson (T) and Tait (T′) themselves, too.

麦克斯韦很快就找到了自己的代号:dp / dt 。这源自泰特 1868 年出版的《热力学概论》一书中热力学第二定律的一种形式:dd=J,其中 JCM 是詹姆斯·克拉克·麦克斯韦的姓名首字母缩写。(我们不需要这个公式,但符号p代表“压力”,J代表“焦耳当量”(热的机械当量),C代表“卡诺函数”,而M与热量有关,今天通常写为dQ / dV。)这很合适,因为麦克斯韦还帮助开创了气体动力学理论——泰特很可能设计了这个方程的形式以表示敬意。麦克斯韦的妻子凯瑟琳曾帮助他进行一些关于热量和颜色的实验,为他点火并调节温度,并进行和记录实验观察。她让我们想起了许多默默无闻的妻子,她们帮助了她们著名的丈夫。3

Maxwell soon picked up a handle of his own: dp/dt. This came courtesy of a form of the second law of thermodynamics in Tait’s 1868 book, Sketch of Thermodynamics: dpdt=JCM, where JCM are James Clerk Maxwell’s initials. (We don’t need this formula, but the symbol p stands for “pressure,” J for “Joule’s equivalent” (the mechanical equivalent of heat), and C for “Carnot’s function,” while M is related to the amount of heat, usually written today as dQ/dV.) It’s fitting, because Maxwell also helped pioneer the kinetic theory of gases—Tait likely crafted this form of the equation as a tribute. Maxwell’s wife, Katherine, had helped with some of his experiments on heat and colour, keeping the fire stoked and regulating the temperature, and making and recording experimental observations. She reminds us of the many unsung wives who have helped their famous husbands.3

虽然TT′中没有四元数和全向量微积分,但有大量分量形式方程。 Tait 一定非常乐意在他的《四元数初等论述》中用很小的篇幅重写了其中一些方程。

While quaternions and whole-vector calculus are missing from T and T′, there are plenty of component-form equations. Tait must have had great delight in rewriting some of them, in a fraction of the space, in his Elementary Treatise on Quaternions.

由于其中一些应用涉及电磁学,他立即建议麦克斯韦研究该书的最后二三十页,了解纳布拉的力量。他的信表明麦克斯韦已经对泰特所说的“∇业务”有了一些了解——事实上,麦克斯韦在 1865 年的论文中用 ∇ 2作为拉普拉斯算子的简写。但现在他正计划写一篇关于电磁学的巨作,概述他建立最终理论的所有已知实验和数学结果,然后打算将其与仍然主导主流物理学的超距作用电磁理论进行比较。当他阅读泰特的书时,他对四元数微积分发展中固有的可能性感到兴奋。

Since some of these applications were to electricity and magnetism, he immediately recommended that Maxwell study the last twenty or thirty pages of the book, on the power of nabla. His letter suggests that Maxwell already had some knowledge of this “∇ business,” as Tait put it—and in fact Maxwell had used ∇2 as shorthand for the Laplacian operator in his 1865 paper. But now he was planning a monumental treatise on electricity and magnetism, surveying all the known experimental and mathematical results on which he’d built his final theory, which he then intended to compare with the action-at-a-distance electromagnetic theories that still dominated mainstream physics. And as he read Tait’s book, he was excited by the possibilities inherent in its development of quaternion calculus.

事实上,当他在《自然》杂志上评论汤姆森和泰特的“巨作”时,他激动不已,不禁感叹书中甚至没有讨论最简单的矢量规则——尤其是因为其中一位作者是“汉密尔顿的忠实信徒”。当然,汤姆森并不是汉密尔顿的忠实信徒!他认为四元数和矢量微积分不过是已经使用分量设计出来的方程的简写——就像麦克斯韦在 1865 年的论文中建立电磁场方程时所做的那样。但现在,在 19 世纪 70 年代初,麦克斯韦发现了矢量微积分,他告诉泰特,汤姆森似乎没有意识到它是“一把燃烧的剑”,贯穿整个空间,而分量形式则是一把强力的笛卡尔“撞锤,只向西、向北和(向下?)推” 。4

In fact, he was so excited that when he reviewed Thomson and Tait’s “great Book” for the journal Nature, he couldn’t help but lament the lack of discussion on even the simplest rules of vectors—especially since one of the authors was “an ardent disciple of Hamilton.” Thomson, of course, was not the ardent disciple! He thought quaternion and vector calculus was nothing but a shorthand for equations that had already been devised using components—as Maxwell had done when he built up his electromagnetic field equations in his 1865 paper. But now, in the early 1870s, Maxwell had discovered vector calculus, and he told Tait that Thomson didn’t seem to realise it was “a flaming sword that burns every way”—all through space— whereas the component form was a brute-force Cartesian “ram, pushing [only] westward and northward and (downward?).”4

麦克斯韦还向他和泰特的儿时好友刘易斯·坎贝尔通报了最新情况,告诉他“我正在转向四元数”,并提到他在撰写《电磁论》时使用了四元数。他解释说,由于运算符 ∇ 被称为“nabla”,因此它的应用是“Nablody”——麦克斯韦喜欢双关语和胡言乱语!泰特同样爱开玩笑,但他也能更专业,他告诉泰特,他“涉猎汉密尔顿”是因为“我想用汉密尔顿的思想充实我的书,但又不想把这些操作转化为汉密尔顿的形式,因为无论是我还是[,]我认为[,]公众都还没有成熟。现在,汉密尔顿的向量思想的价值是无法形容的”——也就是说,好得无法形容! 5他在一篇新论文《物理量分类评论》中解释了他的意思,这篇论文于 1871 年 3 月 9 日提交给伦敦数学学会。(顺便说一句,在 4 月 13 日举行的下一次学会会议上,会长以一篇简短但热情洋溢、当之无愧的悼词拉开了会议的序幕,悼念德摩根——汉密尔顿的老朋友,现代符号代数的先驱——他于 3 月 18 日去世,享年 64 岁。)

Maxwell also kept his and Tait’s childhood friend Lewis Campbell up to date, telling him, “I am getting converted to Quaternions,” and mentioning that he was using them in the Treatise on Electricity and Magnetism he was writing. He explained that since the operator ∇ was called “nabla,” its application was a “Nablody”—Maxwell loved puns and nonsense! With Tait he was equally playful, but he was also able to be much more technical, and he’d told him he was “dabbling in Hamilton” because “I want to leaven my book with Hamiltonian ideas without casting the operations into Hamiltonian form for which neither I nor[,] I think[,] the public are ripe. Now the value of Hamilton’s idea of a vector is unspeakable”—that is, unspeakably good!5 He explained what he meant by this in a new paper, “Remarks on the Classification of Physical Quantities,” which he presented to the London Mathematical Society on March 9, 1871. (Incidentally, at the very next meeting of the society, on April 13, the president opened the proceedings with a brief but glowing and well-deserved elegy for De Morgan—Hamilton’s old friend, and a pioneer of modern symbolic algebra—who had died on March 18, aged sixty-four.)

首先,麦克斯韦谈到了对随着科学进步而产生的大量物理量进行分类的必要性,并指出如果通过数学类比进行分类,那么对于解决方程形式与已知问题相同的新问题将大有裨益。我们之前已经看到,振动弦与麦克斯韦的电磁波之间存在波动方程类比,汤姆逊利用傅立叶热流分析来类比描述法拉第的场线或“力线”,以及麦克斯韦类似的流线类比。麦克斯韦提到了汤姆逊,但他接着说,汉密尔顿通过将物理量分为标量和矢量,提供了一种比类比更基本的分类方法。麦克斯韦说,这一创新成果卓著,四元数的重要性只有笛卡尔发明的坐标系才能与之媲美。他的观点非常正确,因为四元数和向量(及其张量扩展)是现代物理学家、工程师和数据科学家描述和分析物体占据物理和数字空间的方式的基础。回到过去并“见证”一个全新想法(如汉密尔顿的想法)的诞生,并事后观察谁意识到了它的重要性,谁没有意识到,这是一件非常有趣的事情。

First Maxwell spoke of the need to classify the burgeoning number of physical quantities that had arisen because of the progress of science— pointing out that if this classification is done through mathematical analogy, it can be of great help in solving new problems whose equations have the same form as that of a known problem. We saw this earlier with the wave equation analogy between vibrating strings and Maxwell’s electromagnetic waves, and with Thomson’s use of Fourier’s analysis of heat flow as an analogy for describing Faraday’s field lines, or “lines of force,” and with Maxwell’s similar streamline analogy. Maxwell mentioned Thomson, but he went on to say that Hamilton had provided an even more fundamental type of classification than analogy, by dividing physical quantities into scalars and vectors. This innovation was so fruitful, Maxwell said, that the importance of quaternions could only be compared with Descartes’s invention of coordinates. He was spot on, given that quaternions and vectors (and their tensor extensions) are fundamental to the way modern physicists, engineers, and data scientists describe and analyse the way objects occupy physical and digital spaces. It’s fascinating to go back in time and “witness” a brand-new idea, such as Hamilton’s, as it comes into being—and to observe, with hindsight, just who did and who didn’t recognise its significance.

麦克斯韦接着定义了两个新的矢量子类别:力和通量。他说,力是与长度或距离相关的矢量(如线积分),而通量(如磁感应)是与长度或距离相关的矢量到面积或曲面积分。正如我在第 6 章中概述的那样,在他 1865 年的论文中,他使用了散度定理和斯托克斯定理的分量形式将这些积分与导数联系起来。然而,到 1871 年,他发现了哈密顿 nabla 算子 ∇ 的强大功能,它可以用整个矢量和矢量场来定义这些物理量。我们很快就会看到这些 nabla 运算,它们非常重要,麦克斯韦说,他给它们起了特殊的名字。你可能已经知道它们:“散度”、“旋度”和“梯度”——但麦克斯韦把它们给了我们,所以值得花点时间看看他为什么选择它们。

Maxwell went on to define two new subcategories of vectors: forces and fluxes. Forces, he said, are vectors related to length or distance—as in line integrals—and fluxes (such as magnetic induction) are vectors related to areas, or surface integrals. For his 1865 paper he’d used the component form of the divergence theorem and Stokes’s theorem to relate these integrals to derivatives, as I outlined in chapter 6. By 1871, however, he’d discovered the power of Hamilton’s nabla operator, ∇, for defining such physical quantities in terms of whole vectors and vector fields. We’ll see these nabla operations shortly, and they are so important, Maxwell said, that he gave them special names. You might already know them: “divergence,” “curl,” and “grad”—but it was Maxwell who gave them to us, so it’s worth spending a little time seeing why he chose them.

名字的力量

THE POWER OF NAMES

数学的强大之处在于它的视觉象征意义,但要思考你所看到的东西,你还需要能够在头脑中表达出来。麦克斯韦在 1870 年深秋写给泰特的一封信中首次尝试了这个想法:“这里有一些粗略的名字。你能像神一样正确地塑造它们的末端,让它们粘在一起吗?” 6

The great power of mathematics lies in its visual symbolism, but to think about what you see you also need to be able to say it in your head. Maxwell had first tried out the idea in a letter to Tait back in the late autumn of 1870: “Here are some rough hewn names. Will you like a good Divinity shape their ends properly so as to make them stick?”6

首先,他建议,“将 ∇ 应用于标量函数的结果可以称为斜率。”这与斜率类似,=dd,直线y = mx + c的函数,因为当你将 ∇ 应用于标量函数(例如电势(或电压)或力的势)时,同样需要求导。我们通常认为标量就是数字,但如果函数的数值不依赖于你使用的坐标,那么它们就可以是标量。再举一个例子,压力和温度是标量,因此压力和/或温度的函数是标量函数。无论如何,正如我在第 6 章中展示的那样,F = ∇ V(及其现代矢量等价物F = ∇ V)意味着力具有分量类似于直线斜率dd今天我们大声朗读F = ∇ V为“ F等于 grad V ”,其中“grad”是“gradient”的缩写,是“slope”的另一个名称。所以,麦克斯韦的术语确实有点道理!它显示了标量函数在空间中的变化方式。

First up, he suggested, “the result of ∇ applied to a scalar function might be called the slope.” That’s by analogy with the slope, m=dydx, of a straight line y = mx + c, because similarly you’re taking derivatives when you apply ∇ to a scalar function, such as the electric potential (or voltage), or the potential of a force. We often think of scalars as just numbers, but functions can be scalars if they take numerical values that don’t depend on the coordinates you’re using. To take another example, pressure and temperature are scalars, so a function of pressure and/or temperature is a scalar function. Anyway, as I showed in chapter 6, F = ∇V (and its modern vector equivalent F = ∇V ) means that the force has components Vx,Vy,Vz, analogous to the straight-line slope dydx. Today we read F = ∇V aloud as “F equals grad V,” where “grad” is short for “gradient,” which is another name for “slope.” So, Maxwell’s term did sort of stick! It shows how a scalar function changes through space.

接下来,麦克斯韦谈到了将 ∇ 应用于向量时会发生什么。我通常使用粗体表示向量,以方便现代读者阅读,但正如我所指出的,四元数形式本质上是一样的:现代向量基和汉密尔顿虚数基之间的差异只有在标量积中才会显现出来。这是因为根据虚数的定义,i 2 = −1,而ii定义等于 1,jk也是如此。(至少在我们遇到 Heaviside 时,它​​们很快就会被这样定义。)因此,每当你取两个向量的标量积时,例如

Next, Maxwell spoke of what happens when you applied ∇ to vectors. I’ve generally been using bold type for vectors to make it easier for modern readers, but as I’ve noted it’s essentially the same in quaternion form: the difference between the modern vector basis and Hamilton’s imaginary number basis only becomes apparent in the scalar product. That’s because i2 = −1, by the definition of an imaginary number, whereas ii is defined to equal 1, and similarly for j and k. (At least they will be so defined shortly, when we meet Heaviside.) So, whenever you take the scalar product of two vectors, such as

v = v 1 i + v 2 j + v 3 kw = w 1 i + w 2 j + w 3 k

v = v1i + v2 j + v3k and w = w1i + w2 j + w3k

在汉密尔顿符号中,你也要乘以基础量。这意味着使用汉密尔顿符号表示标量积(泰特和麦克斯韦也使用过),你有

in Hamilton’s notation, you’re multiplying the basis quantities too. Which means that using Hamilton’s notation for scalar products, which Tait and Maxwell used, too, you have

S. vw = v 1 w 1 i 2 + v 2 w 2 j 2 + v 3 w 3 k 2 = −( v 1 w 1 + v 2 w 2 + v 3 w 3 )。

S. vw = v1w1i2 + v2w2 j2 + v3w3k2 = −(v1w1 + v2w2 + v3w3).

因此,汉密尔顿的标量积和现代的标量积之间的唯一区别就是前面有一个减号:S.vw = −v w 类似地,nabla 算符和v的标积是S.∇v = −∇∙ v 。麦克斯韦建议我们将其读作矢量场v的“收敛” ,但你可能会认出∇∙ v是v的“发散” ——这个名字是后来才引入的,因为当你去掉减号时,矢量会发散而不是收敛(如图 7.1所示)。所以在这里,麦克斯韦的名字仍然保留了下来。

So, the only difference between Hamilton’s scalar product and the modern one is that minus sign out front: S. vw = −vw. Similarly, the scalar product of the nabla operator and v is S. ∇v = −∇ ∙ v. Maxwell suggested we read this as the “convergence” of the vector field v, but you might recognise ∇ ∙ v as the “divergence” of v—a name introduced later, because when you lose the minus sign, your vectors diverge instead of converging (as in fig. 7.1). So here again, Maxwell’s name did essentially stick.

在他的《电子论》中,麦克斯韦给出了图 7.1b所示的图表来说明收敛。他用这个想法来证明(现在所说的)“作用在测试电荷上的电力”——他称之为“电动势强度”,𝔈,现在称为“电场”,写成E——的散度等于 4π 乘以电荷密度 ρ。(正如他在纳布罗迪信中告诉坎贝尔的那样,他已经用完了希腊字母(汉密尔顿选择的矢量),并且“异端地”使用德语大写字母,例如 𝔈。)麦克斯韦以图 7.1 标题中给出的分量形式写出了这个方程但他还给出了一个全矢量版本,忽略汉密尔顿标量积中烦人的减号,他写成这样:

In his Treatise, Maxwell gave the diagram shown in figure 7.1b to illustrate convergence. He used the idea to show that (what is now called) the divergence of “the electric force on a test charge”—which he called the “electromotive intensity,” 𝔈, and which is now called the “electric field,” written as E—is equal to 4π times the electric charge density, ρ. (As he told Campbell in his Nablody letter, he’d run out of Greek letters, Hamilton’s choice for vectors, and was “heretically” using German capitals such as 𝔈 instead.) Maxwell wrote this equation in the component form given in the caption to figure 7.1. But he also gave a whole-vector version, which, ignoring the annoying minus sign in Hamilton’s scalar product, he wrote like this:

ρ = S.∇𝔇

ρ = S. ∇𝔇.

图像

图 7.1。发散。例如,库仑定律告诉我们,如果P处有一个带正电的粒子,而你在P周围的场中各点放置一个单位正测试电荷,则电力的方向如 (a) 中的箭头所示。当你远离P时,力会根据平方反比定律而减小,因此箭头会变小。如果测试电荷为负,它会被P处的正电荷吸引,因此会出现情况 (b)。在《电磁论》第 1 卷(第 77 条)中,麦克斯韦利用高斯关于库仑定律与通量相关的思想,将该定律写为

FIGURE 7.1. Divergence. For example, Coulomb’s law tells us that if there’s a positively charged particle at P, and you place a unit positive test charge at various points in the field around P, the electric force is in the direction shown by the arrows in (a). As you move away from P, the force drops off according to the inverse square law, so the arrows would get smaller. If the test charge is negative, it is attracted to the positive charge at P, so you have case (b). In volume 1 (art. 77) of his Treatise on Electricity and Magnetism, using Gauss’s idea that Coulomb’s law is related to flux, Maxwell wrote this law as

++=4πρ,

Xx+Yy+Zz=4πρ,

等式左边是电力的散度(他将其分量写为X、Y、Z),ρ 是场的电荷密度。我在第 6 章的尾注中展示了麦克斯韦如何利用通量的概念推导出这个方程。下图 (c) 看起来像条形磁铁周围的力线,如图 6.3a所示——我们将会看到,麦克斯韦方程的另一个特点是磁场的散度为零。

where the left-hand side of the equation is the divergence of the electric force (whose components he wrote as X, Y, Z), and ρ is the electric charge density of the field. I showed how Maxwell used the idea of flux to derive this equation in an endnote to chapter 6. The lower diagram in (c) looks like the lines of force around a bar magnet, shown in figure 6.3a—and as we’ll see, another of Maxwell’s equations is that the divergence of the magnetic field is zero.

因为他已经定义了 𝔇 = 𝔈/4π,所以这只是

Since he’d already defined 𝔇 = 𝔈/4π, this is just

4πρ = ∇ ∙ E

4πρ = ∇ ∙ E

用现代符号表示。(𝔇 是麦克斯韦所谓的“电位移”,但它是我们这里需要的方程,而不是物理细节。)7

in modern notation. (𝔇 is Maxwell’s so-called “electric displacement,” but it’s the equations we need here, not the physical details.)7

我们很快就会看到这种现代符号转换是如何发生的,以及为什么会发生,但现代读者可能已经对矢量微积分有所了解,有些人也知道麦克斯韦方程——所以我试图将历史过程与你可能已经知道的内容联系起来。然而,关键是,在每一种符号形式中,引人注目的是物理量:电荷密度和发散电场。相比之下,当你以图 7.1所示的分量形式写出同样的方程时,重点在于数学而不是物理。这就是为什么麦克斯韦写信给泰特说,与汤姆森等“不信教者”的观点相反,“4nion(他对四元数的简称)的优点目前并不在于解决难题,而在于让我们看到问题的意义及其解决方案。” 8

We’ll see shortly just how and why this modern transformation in notation happened, but modern readers will likely have some knowledge already of vector calculus, and some will know Maxwell’s equations, too—so I’m trying to relate the historical process to what you may already know. The key thing, though, is that in each of these notational forms, the striking thing is the physical quantities: the charge density, and the diverging electric field. By contrast, when you write the same equation in the component form shown in figure 7.1, the emphasis is on the maths rather than the physics. That’s why Maxwell wrote to Tait saying that contrary to the view of “unbelievers” such as Thomson, “the virtue of the 4nions [his shorthand for quaternions] lies not so much as yet in solving hard questions, as in enabling us to see the meaning of the question and its solution.”8

麦克斯韦用符号𝔅表示他所谓的磁感应;今天我们通常简称它为磁场矢量B。对于静止磁铁,所有磁力线都始于磁铁的北极,终于磁铁的南极,如图7.1c所示,因此在磁铁周围的封闭表面没有净磁通量——这意味着磁场矢量B没有发散。(为了看到这一点,想象一个被包裹在一个大球中的条形磁铁。磁力线从磁铁的一端绕到另一端——它们不会单独分支并单向穿过球的表面,就像它们穿过图 6.2中电动机/发电机中的旋转环一样。)换句话说,与电子或质子等带电粒子(要么是正极要么是负极)不同,天然磁铁总是有两个极。没有人检测到自然产生的磁单极子,尽管它们已经在实验室的虚拟量子水平上被创造出来;它们也从理论上出现在弦理论中,这是保罗·狄拉克在 1931 年的工作成果。(我和我的同事托尼·伦是长期探索引力磁单极子存在的人之一。这是利用不同主题之间的相似性进行类比的另一个例子。9

Maxwell used the symbol 𝔅 for what he called the magnetic induction; today we often call this simply the magnetic field vector, B. For a stationary magnet, all the field lines begin at the magnet’s north pole and end at its south, as in figure 7.1c, so there’s no net magnetic flux through a closed surface around the magnet—which means there’s no divergence of the magnetic field vector B. (To see this, imagine the bar magnet enclosed in a large ball. The field lines loop from one end of the magnet to the other—they don’t branch out separately and pass one-way through the ball’s surface, the way they do through the rotating loop in the electric motor/generator in fig. 6.2.) This is another way of saying that, unlike a charged particle such as an electron or proton, which is either negative or positive, natural magnets always have two poles. No one has detected a naturally occurring magnetic monopole, although they’ve been created in the lab at the virtual quantum level; they also arise theoretically in string theory, following from the work of Paul Dirac in 1931. (My colleague Tony Lun and I are among those who have long been exploring the existence of gravitomagnetic monopoles. It’s another example of the use of analogies between different topics because of similarities in the equations.9)

麦克斯韦以分量形式给出了磁散度方程,而不是明确指出散度(或收敛度),即nabla 和 𝔅 的标量积,等于零:S . ∇𝔅 = 0,或用现代符号表示为 ∇ ∙ B = 0。但他确实给出了一个等效的全矢量形式方程,我们稍后会看到。10不过,他首先必须为nabla 的矢量积找到一个名称。

Maxwell gave his magnetic divergence equation in component form— rather than specifically saying that the divergence (or convergence), the scalar product of nabla and 𝔅, equals zero: S. ∇𝔅 = 0, or ∇ ∙ B = 0 in modern notation. But he did give an equivalent equation in whole-vector form, which we’ll see in a minute.10 First, though, he had to find a name for the vector product with nabla.

正如我们在图 4.1中看到的向量积的右手定则,如果你将手指从一个向量弯曲到另一个向量,那么你的拇指指向的是它们向量(或交叉)积的方向。它值得注意的是,这种卷曲运动也具有物理意义,如果用手指表示刚体绕着通过拇指的固定轴旋转的运动,那么 nabla 与刚体上任意点的速度的矢量积都沿着同一条轴(如图 7.2所示)。所以,nabla 的矢量积与旋转有关。在给泰特的信中,麦克斯韦为这种操作提出了几个可能的名字,包括“旋转”——他认为这个名字“足够活泼”,虽然“对纯数学家来说可能太动态了”。所以,“看在凯莱的份上”,他说,“卷曲”这个名字怎么样?凯莱发展了“卷轴”或“斜曲面”的数学——就像你在弦雕中看到的扭曲的直纹曲面——当然,当你卷起卷轴时,你会“卷曲”它。在他的《数学论文》第 25 和 26 篇文章中,麦克斯韦将 ∇ × 运算命名为“旋度”,尽管“非常犹豫”——这个名字也沿用了下来。

As we saw in the right-hand rule for vector products in figure 4.1, if you curl your fingers in the direction from one vector to another, your thumb points in the direction of their vector (or cross) product. It turns out that this curling motion is also physically significant, in the sense that if your fingers indicate the motion of a rigid body rotating around a fixed axis through your thumb, the vector product of nabla and the velocity of any point on the body lies along this same axis (as in fig. 7.2). So, the vector product with nabla has something to do with rotations. In his letter to Tait, Maxwell suggested several possible names for this operation, including “twirl”—which, he thought, was “sufficiently racy,” although perhaps “too dynamical for pure mathematicians.” So, “for Cayley’s sake,” he said, what about the name “curl.” Cayley had developed the mathematics of “scrolls” or “skew surfaces”—like the twisted ruled surfaces you might have seen as string sculptures—and, of course, you “curl” a scroll when you roll it up. In articles 25 and 26 of his Treatise, Maxwell settled on the name “curl” for the operation ∇ ×, albeit “with great diffidence”—and this name, too, stuck.

图像

图 7.2 . 旋转圆盘在任意点P处的速度旋度位于所示方向。

FIGURE 7.2. The curl of the velocity of the rotating disk at any point P lies in the direction shown.

在他的电磁场方程中,麦克斯韦这样表达了他的一个旋度运算:

In his electromagnetic field equations Maxwell expressed one of his curl operations like this:

𝔅 = V . ∇𝔘

(现代符号为B = ∇ × A ),

𝔅 = V. ∇𝔘

(B = ∇ × A in modern notation),

其中,他的 𝔘 和现代的A代表磁场B的矢量势。(第 6 章中描述的标量势具有物理解释,与功和势能有关——一个例子是电压。然而,电磁场的矢量势本质上是一个数学量,它通过斯托克斯定理与磁场相关。)11你可能从数学课上知道——正如麦克斯韦从 Tait 的工作中知道的那样——旋度的散度始终为零。所以,这个方程实际上等同于S。∇𝔅 = 0(或∇∙ B = 0)。12

where his 𝔘 and the modern A represent the vector potential of the magnetic field B. (The scalar potential described in chap. 6 has a physical interpretation, related to work and potential energy—and an example is the voltage. The vector potential of the electromagnetic field, however, is essentially a mathematical quantity that is related to the magnetic field via Stokes’s theorem.)11 As you may know from maths class—and as Maxwell knew from Tait’s work—the divergence of a curl is always zero. So, this equation was, in fact, equivalent to S. ∇𝔅 = 0 (or ∇ ∙ B = 0).12

麦克斯韦在《人性论》中还提出了其他旋度方程,它们共同构成了麦克斯韦方程现代形式的著名旋度方程,我们将在下一章中看到。现在让我们暂停一下,因为事后可能很难意识到麦克斯韦展示如何以全矢量形式写出电磁理论时到底是一个多大的突破。它对我们的故事的影响将是巨大的,尤其是对物理学——因此,对我们理解宇宙的方式也是如此。

Maxwell also had other curl equations in the Treatise, and together they led to the famous curl equations in the modern form of Maxwell’s equations that we’ll see in the next chapter. Meantime let’s pause for a moment, for it might be hard to appreciate in hindsight just what a breakthrough it was when Maxwell showed how to write his theory of electromagnetism in whole-vector form. Its ramifications for our story will be huge, especially for physics—and, therefore, for the way we understand our universe.

整体载体的力量

THE POWER OF WHOLE VECTORS

1872 年 10 月 19 日,麦克斯韦在检查《人性论》的校样时,向他的老朋友坎贝尔解释说,他的出版商克拉伦登出版社在 13 周内平均完成了 9 张纸。这再次提醒我们当时的技术,当时书的每一页都必须手工排版,一次排版一个字母。至于校对,这总是很费力,但麦克斯韦告诉坎贝尔,他们共同的朋友泰特“在发现荒谬之处方面给了我很大的帮助。” 13

On October 19, 1872, while he was checking the proofs of his Treatise, Maxwell explained to his old friend Campbell that his publisher, Clarendon Press, managed to average around nine sheets in thirteen weeks. It’s another reminder of the technology of the times, when each page in a book had to be typeset manually, one painstaking letter at a time. As for proofreading, it is always laborious, but Maxwell told Campbell that their mutual friend Tait “gives me great help in detecting absurdities.”13

1873 年,麦克斯韦的《电磁论》终于问世。同年,他在伦敦皇家学会举行了一场特别演讲,以纪念詹姆斯·瓦特——这位十八世纪的苏格兰工程师使最初的纽科门蒸汽机效率大大提高。托马斯·纽科门设计的蒸汽机用于从矿井中抽水,但瓦特改进后的蒸汽机应用范围更广,为工业革命提供了动力。麦克斯韦谈到,由于所有这些新兴产业,我们的社会和就业互动日益复杂,并敦促听众在“机器的轰鸣声和商业的压力中”记住生活中更简单、更“高尚”的方面。14

Maxwell’s Treatise on Electricity and Magnetism finally appeared in 1873. In the same year, he gave a special lecture at London’s Royal Institution, in honour of James Watt—the eighteenth-century Scottish engineer who made the original Newcomen steam engine so much more efficient. Thomas Newcomen had designed his engine to pump water out of mines, but Watt’s improved engines had much wider application, helping power the Industrial Revolution. Maxwell spoke of the increasing complexity of our social and employment interactions because of all this new industry, and urged his listeners to keep in mind, “amid the rattle of machinery and the press of business,” the simpler, more “noble” aspects of life.14

图像

詹姆斯·克拉克·麦克斯韦的雕像,2009 年在爱丁堡竖立起来。他忠实的狗托比躺在他的脚边,他手里拿着色轮,正是色轮帮助他理解了彩色视觉和彩色摄影的本质。

Statue of James Clerk Maxwell, erected in Edinburgh in 2009. His faithful dog Toby lies at his feet, and he’s holding the colour wheel that helped him understand the nature of colour vision and colour photography.

至于整个向量的优点,在 1873 年《自然》杂志对 Tait 关于四元数的新书的评论中,麦克斯韦指出,虽然计算对于数学工作至关重要,但数学家不仅仅是一个计算器(或牛顿所说的“枯燥乏味的苦差事”);相反,富有创造力的数学家发明了新的、节省劳动力的方法。但是,尽管汤姆森认为四元数和向量只是已经解出的方程的简写,但麦克斯韦认为“四元数或向量学说一种数学方法,但它是一种思维方法”——它不仅仅是一种创造性的节省劳动力的计算手段。15

As for the virtues of whole vectors, in an 1873 Nature review of Tait’s new book about quaternions, Maxwell pointed out that while calculations are vital to the job of mathematics, a mathematician is much more than a mere calculator (or “dry drudge,” as Newton had put it); rather, creative mathematicians invent new, labour saving methods. But while Thomson believed quaternions and vectors were just a shorthand for equations already worked out, Maxwell saw that “Quaternions, or the doctrine of Vectors, is a mathematical method, but it is a method of thinking”—it is not just a creative labour saving device for calculations.15

1870 年,泰特因两篇关于四元数的论文荣获爱丁堡皇家学会基思奖章,其中一篇是关于物体绕固定旋转,而不是绕轴旋转,如图 7.2所示。 麦克斯韦也曾以诙谐幽默的方式表达过同样的观点。(这是一个难题,19 世纪 80 年代末,女数学家先驱索尼娅·科瓦列夫斯基因对此类运动的分析而声名鹊起,并因此获奖。与泰特不同,她使用的是分量方法,而不是矢量微积分方法,因为矢量微积分方法的实用性当时仍存在争议。)在纪念泰特的演讲中,麦克斯韦说,数学家经常会因为计算而疲惫不堪,没有精力思考,而“泰特却能让他通过思考来思考,这是一种更高尚但更昂贵的职业,而且他不会像要解几页方程式那样犯那么多错误。”泰特对麦克斯韦的讽刺之语很感兴趣!16

He’d made the same point, with fey humour, when Tait was awarded the Royal Society of Edinburgh’s Keith Medal in 1870, for two of his quaternion papers—including one on the rotation of a body around a fixed point rather than around an axis as in figure 7.2. (It was a difficult topic, and the pioneering female mathematician Sonia Kovalevsky would become famous in the late 1880s for her own award-winning analysis of such motions. Unlike Tait, she used components, not vector calculus methods, for their usefulness was still being contested.) In his speech in honour of Tait, Maxwell said that a mathematician often gets so tired with all his calculating that he has no energy for thinking—whereas “Tait is the man to enable him to do it by thinking, a nobler though more expensive occupation, and in a way by which he will not make so many mistakes as if he had pages of equations to work out.” Tait was delighted by Maxwell’s wry take!16

在《数学与系统论》的开篇,麦克斯韦对此进行了更为简洁的阐述,他说,在“物理推理”中,而不是在计算中,你希望看到的是你所研究的对象本身,而不是关注它的坐标和组成部分。他在出版物和信件中反复强调了这一点——通过整个向量,你可以看到你正在做的事情的物理意义,而不是迷失在计算中。17然而,除了泰特之外,似乎很少有人听。

At the beginning of his Treatise, Maxwell put this more succinctly, saying that in “physical reasoning” rather than calculating, you want to see the object you’re studying for itself, as a whole, rather than focussing on its coordinates and components. He made this point over and over, in his publications and his letters—that with whole vectors you can see the physical meaning of what you’re doing, rather than getting lost in the calculations.17 Aside from Tait, though, few seemed to be listening.

汤姆森也许是最著名的反对者。他甚至不接受麦克斯韦的电磁学理论,认为它就像神秘主义——全是数学,没有机械模型具体解释电磁波是如何传播的!汤姆森虽然才华横溢,也承认麦克斯韦广泛的科学天赋,但他从未意识到数学“矢量场”的力量,而这种力量如今已成为物理学的基础,也已应用于生物学和其他科学领域。即使在 19 世纪 90 年代,海因里希·赫兹制造出证实麦克斯韦理论的无线电波后,汤姆森仍建议同事不要使用“矢量”一词。他说,它对问题几何的简单性毫无贡献,并奇怪地补充道:“四元数是汉密尔顿在完成真正出色的工作后提出的;虽然非常巧妙,但对那些以任何方式接触过它们的人来说,包括克拉克·麦克斯韦,它都是纯粹的邪恶。” 18

Thomson was perhaps the most famous dissenter. He didn’t even accept Maxwell’s theory of electromagnetism in component form, likening it to mysticism—all that maths, with no mechanical model explaining in concrete terms how electromagnetic waves propagate! Brilliant as he was—and much as he recognised Maxwell’s wide-ranging scientific genius— Thomson never realised the power of the mathematical “vector field” that now underlies so much of physics, and which has also found applications in biology and other sciences. Even in the 1890s, after Heinrich Hertz had produced the radio waves that confirmed Maxwell’s theory, Thomson advised a colleague against using the term “vector.” It contributes nothing, he said, to the simplicity of the geometry of a problem, adding bizarrely, “Quaternions came from Hamilton after his really good work had been done; and, though beautifully ingenious, have been an unmixed evil to those who have touched them in any way, including Clerk Maxwell.”18

19 世纪 90 年代,斯托克斯也拒绝教授麦克斯韦理论,年轻的爱因斯坦的教授们也拒绝了——部分原因是由于赫兹和其他德语人士的影响,该理论才刚刚开始在德国崭露头角;因此爱因斯坦逃学以便自己研究它。几年后,他将麦克斯韦方程融入了他著名的 1905 年狭义相对论中——他谈到了矢量,尽管他只使用了分量形式。然而十年后,他在介绍广义相对论的论文中以现代全矢量形式和张量形式写出了麦克斯韦方程。

Stokes, too, refused to teach Maxwell’s theory in the 1890s, and young Einstein’s professors also refused—partly because the theory had only recently begun to make its mark in Germany, thanks to Hertz and other German-speakers; so Einstein played truant in order to study it for himself. In a few years’ time, he would incorporate Maxwell’s equations into his famous 1905 theory of special relativity—and he’d speak of vectors, although he’d use only component forms. A decade later, however, he would write Maxwell’s equation in both modern whole-vector form and tensor form, in his paper introducing the general theory of relativity.

我们稍后会看到更多爱因斯坦的作品。这里的重点是,麦克斯韦的电磁理论和他对矢量微积分的赞扬花了很长时间才得到广泛接受。他于 1865 年发表了该理论,并于 1873 年在《人性论》中以矢量微积分的形式发表了该理论,但他一定觉得没有人真正关心它。

We’ll see more of Einstein’s work later. The point here is that it took a long time for Maxwell’s theory of electromagnetism, and his praise for vector calculus, to find widespread acceptance. He published the theory in 1865, and in vector-calculus form in the Treatise in 1873, but it must have seemed to him that no one really cared.

形势开始逆转

THE TIDE BEGINS TO TURN

1873 年,麦克斯韦(和泰特) 42 岁,而崭露头角的威廉·金登·克利福德 28 岁。克利福德和汉密尔顿一样,也是神童。他以第二名的成绩从剑桥大学毕业:和麦克斯韦一样,他更喜欢学习自己感兴趣的东西,而不是为了获得大学荣誉而临时抱佛脚——他对亚瑟·凯莱和詹姆斯·西尔维斯特的前沿代数工作特别感兴趣。和麦克斯韦一样,克利福德也是一名运动员:麦克斯韦是一名出色的骑手,也是一名强壮但不太优雅的游泳运动员(而泰特更喜欢打高尔夫球),但克利福德是一个力大无穷的人——他只用一只手臂就能在体操运动员的单杠上站起来。他知道麦克斯韦和泰特的工作,三人在英国科学促进会和伦敦和爱丁堡皇家学会的各种会议上相识,成为了朋友。19

In 1873, Maxwell (and Tait) turned forty-two—and the up-and-coming William Kingdon Clifford turned twenty-eight. Clifford had been a child prodigy like Hamilton. He graduated Second Wrangler from Cambridge: like Maxwell, he’d preferred to study what interested him rather than cramming for the Tripos—he was especially interested in the cutting-edge algebraic work of Arthur Cayley and James Sylvester. Also like Maxwell, Clifford was quite an athlete: Maxwell was a superb horseman and a strong if inelegant swimmer (Tait, on the other hand, preferred golf ), but Clifford was a man of prodigious strength—he could pull himself up on a gymnast’s bar with just one arm. He knew of Maxwell and Tait’s work, and the three men became friendly, having met at various meetings of the British Association for the Advancement of Science and the Royal Societies of London and Edinburgh.19

1877 年,克利福德在伦敦大学学院(德·摩根曾任教的世俗大学)举办了一系列关于四元数的讲座,以抗议牛津大学和剑桥大学职位所需的圣公会誓言。同年,克利福德在当时围绕科学和宗教的激烈辩论中采取了有争议的立场,他写道无神论文章认为,我们需要确凿的证据才能相信某件事——光有信仰是不够的。20

In 1877 Clifford gave a series of lectures on quaternions at University College London—the secular university where De Morgan had taught, in protest at the Anglican vows needed for a position at Oxford and Cambridge. In the same year, Clifford took a controversial stand in the fierce debates around science and religion that were then taking place, writing atheistic articles suggesting that we need firm evidence before we believe something—faith alone was not enough.20

1878 年,他在其著作《动力学要素》中对向量代数给出了自己的数学处理:尽管他是一位纯数学家,但他强调了向量在动力学(即研究力引起的运动的学科)中的重要性。(在此过程中,他用现代术语“发散”取代了麦克斯韦的负数“收敛”。)在评论克利福德的著作时,泰特称赞麦克斯韦的“伟大”著作《四元数论》可能是除专业四元数文献之外,第一部推动向量方法在物理学中应用的著作——这也是我花了大量时间研究麦克斯韦的原因,因为他的《四元数论》是向量历史的转折点。泰特接着表示,克利福德“将这项优秀的工作发扬光大,(如果仅仅因为这个原因),我们希望他的书会受到广泛欢迎。” Tait 的幽默感与 Maxwell 一样调皮,他借此抨击 Clifford 将“Dynamic”写成了“Dynamics”;更严肃的是,他为 Clifford 引入了太多其他奇特且不必要的新术语而感到遗憾。但他尤其遗憾的是 Clifford 没有充分利用 nabla 算子:就好像他在四元数的道路上走了很长一段路,然后突然停了下来。21

In 1878, he gave his own mathematical treatment of vector algebra in his book Elements of Dynamic: although he was a pure mathematician, he emphasised the importance of vectors in dynamics, the study of motion due to forces. (In the process, he replaced Maxwell’s negative “convergence” with our modern term “divergence.”) In his review of Clifford’s book, Tait praised Maxwell’s “great” Treatise as perhaps the first, outside the specialist quaternion literature, to promote the use of vector methods in physics—which is why I’ve spent quite a bit of time on Maxwell, for his Treatise is a turning point in the story of vectors. Tait went on to say that Clifford “carries the good work a great deal farther, and (if for this reason alone), we hope his book will be widely welcomed.” Tait had as mischievous a sense of humour as Maxwell, and he used it to take a swipe at Clifford’s use of “Dynamic” instead of “Dynamics”; more seriously, he lamented Clifford’s introduction of too many other idiosyncratic and unnecessary new terms. But he especially regretted that Clifford hadn’t made sufficient use of the nabla operator: it was as if he’d gone quite a way along the quaternionic road and then suddenly stopped.21

尽管如此,在 19 世纪 70 年代的各种著作中,克利福德做出了一些特别的贡献,这些贡献不仅使他成为汉密尔顿、泰特和麦克斯韦的继承者,也使他成为格拉斯曼的继承者。他是极少数仔细研究汉密尔顿格拉斯曼的人之一,他着手创建一种将这两种方法结合起来的新代数。他称之为“几何代数”,并以“几何积”为基础,将汉密尔顿的标量积(格拉斯曼的内积)和格拉斯曼的外积(汉密尔顿三维矢量积的更通用版本)结合起来。克利福德的几何积在现代符号中看起来像这样(但与麦克斯韦和泰特一样,克利福德使用汉密尔顿的SV表示标量和矢量积,他们都没有使用现代粗体表示矢量):

Still, in his various writings through the 1870s Clifford had done something special, something that made him a worthy successor not only to Hamilton, Tait, and Maxwell, but to Grassmann, too. He was one of the very few who had closely studied both Hamilton and Grassmann, and he set out to create a new algebra that united the two approaches. He called it a “geometric algebra,” and he based it around a “geometric product” that united Hamilton’s scalar product (Grassmann’s inner product) and Grassmann’s outer product (which was a more general version of Hamilton’s three-dimensional vector product). Clifford’s geometric product looks like this in modern notation (but like Maxwell and Tait, Clifford used Hamilton’s S and V for scalar and vector products, and none of them used modern boldface for vectors):

ab = a b + a b

ab = ab + ab,

其中 ∧ 被称为“楔形”——在三维空间中,ab等价于a × b 的叉积。汉密尔顿的全四元数积也是这两类矢量乘法的组合,正如我在第 4 章中展示的那样(wa分别是四元数PQ中的标量,pq是矢量部分):

where ∧ is called a “wedge”—in three dimensions ab is equivalent to the cross product a × b. Hamilton’s full quaternion product is also a combination of these two types of vector multiplication, as I showed in chapter 4 (w and a are the scalars in the quaternions P and Q , respectively, and p and q are the vector parts):

PQ = wapq + wq + ap + p × q

PQ = wapq + wq + ap + p × q.

但它只适用于三维空间。

But it applies only in three dimensions.

对于日常三维世界中的物理学,克利福德和麦克斯韦一样发现,单独的标量和矢量积比完整的四元数积更有用——这是从四元数向现代矢量分析迈出的又一步。但克利福德本质上是一位纯数学家,纯数学家喜欢尽可能地推动数学的发展,不仅是为了好玩——也是为了未来的应用。对于这样的数学家来说,四元数的有趣之处在于,除了它们的乘积不交换之外,它们还遵循所有常见的代数定律——结合律、分配律等等——而且,就像在普通代数中一样,你也可以用它们进行除法。用数学术语来说,这意味着四元数q有一个“逆” q −1,定义为qq −1 = 1,其中 1 是四元数,1 + 0 i + 0 j + 0 k。22例如,在表示旋转的四元数框架中,逆会反转旋转的方向。如果你想解方程,逆矩阵也很重要。你可能从矩阵方程中熟悉这个想法,例如AX = B ,其中如果矩阵A有逆矩阵,则解为X = A −1 B 。

For physics in the everyday 3-D world, Clifford, like Maxwell, found that the separate scalar and vector products were far more useful than the full quaternion product—and this was another step toward the break away from quaternions to modern vector analysis. But Clifford was a pure mathematician at heart, and pure mathematicians love to push the maths as far as they can, for the fun of it—as well as for inklings of future applications. For such a mathematician, the interesting thing about quaternions is that aside from the fact that their product isn’t commutative, they obey all the usual laws of algebra—associative, distributive, and so on—and, as in ordinary algebra, you can divide with them, too. In mathematical terms this means a quaternion q has an “inverse” q−1, defined so that qq−1 = 1, where the 1 here is a quaternion, 1 + 0i + 0j + 0k.22 In the quaternion framework for representing rotations, for example, the inverse reverses the direction of the rotation. Inverses are also important if you want to solve equations. You might be familiar with this idea from matrix equations such as AX = B, where the solution is X = A−1B, if the matrix A does have an inverse.

但问题是:仅使用向量积是无法求逆的——你需要完整的四元数积。同样,Clifford 的完整几何积也有一个逆。正如四元数在计算三维空间中的复杂旋转时被证明是如此有效一样,现代数学家已经意识到几何积可以更有效地完成许多使用向量可以做的事情。23

But here’s the thing: there’s no such inverse using just vector products— you need the full quaternion product. Similarly, Clifford’s full geometric product has an inverse. And just as quaternions belatedly proved so efficient for calculating complex rotations in three-dimensional space, so modern mathematicians have realised that the geometric product enables many of the things you can do with vectors to be done more efficiently.23

令人悲伤的是,被低估的格拉斯曼也曾试图统一他的体系和汉密尔顿的体系——1877 年,也就是他 68 岁完全离开人世的那一年。他虽然离开了,但并没有被遗忘,因为克利福德是第一个展现自己才华的人,他独创了将两个系统统一起来的方法。当时也没有多少人关注克利福德。

In a sad irony, the undervalued Grassmann had also been trying to unify his system and Hamilton’s—in 1877, the very year he slipped out of the world altogether, at the age of sixty-eight. He was gone but not forgotten, as Clifford was the first to show, with his own unification of the two systems. Not that many people back then were paying attention to Clifford, either.

十年后,意大利数学家朱塞佩·皮亚诺也发现了格拉斯曼的《脱离论》,并发表了自己对它的总结和扩展——包括第一个现代的、公理化的向量空间定义。这使向量的概念超越了箭头或分量列表等直观模型,而是对其进行了抽象定义——作为向量空间的元素,其结构通过类比实数来定义。正如我们在汉密尔顿和他的朋友奥古斯都·德·摩根早期关于代数定律的工作中看到的那样,这种结构意味着在将向量相加和将其乘以标量时,闭包律、交换律、分配律和结合律成立,并且存在恒等元和加法逆元。不过,我们的故事不需要细节——我们只需要这样一个想法:向量比那些学校水平的箭头更抽象。此外,也没有多少人听皮亚诺的课。但他和克利福德的工作体现了格拉斯曼死后命运的变化,随着数学家们寻求更为严格的定义,这种变化开始发生。

A decade later, the Italian mathematician Giuseppe Peano would also discover Grassmann’s Ausdehnungslehre, and he would publish his own summary and extension of it—including the first modern, axiomatic definition of a vector space. This takes the idea of vectors beyond intuitive models such as arrows or lists of components, defining them abstractly instead—as elements of a vector space whose structure is defined by analogy with that of real numbers. As we saw with the earlier work of Hamilton and his friend Augustus De Morgan on the laws of algebra, this structure means that when adding vectors, and when multiplying them by a scalar, the closure, commutative, distributive, and associative laws hold, and identity and additive inverse elements exist. We don’t need the details for our story, though—we just need the idea that vectors are more abstract than those school-level arrows. Besides, few were listening to Peano, either. But his and Clifford’s work exemplified the change in Grassmann’s posthumous fortune that was beginning to take place as mathematicians sought ever more rigorous definitions.

跨文化插曲

A CROSS-CULTURAL INTERLUDE

除了四元数之外,克利福德、泰特和麦克斯韦之间还有一些有趣的松散联系,这有助于将他们的工作和时代置于更广泛的科学和文化背景中。

Aside from quaternions, there are some interesting loose connections between Clifford, Tait, and Maxwell, which help put their work and times into a broader scientific and cultural context.

克利福德是著名小说家兼评论家乔治·艾略特及其伴侣乔治·刘易斯的朋友——克利福德和他的文学妻子露西经常参加他们著名的周日下午聚会,聚会上有作家、科学家、哲学家和感兴趣的朋友。和刘易斯一样,艾略特也对科学和数学感兴趣。19 世纪 50 年代,她曾在一个“为女士”开设的成人班学习几何——1882 年,泰特向凯莱抱怨说,他“几乎被迫”为“女士”开设一门课程,但问题可能出在他的工作量上,而不是女人。现在,在 19 世纪 70 年代,艾略特正在阅读新的非欧几里得几何学,我们将在第 10 章中看到更多内容,克利福德也在其中留下了自己的印记。24

Clifford was a friend of the venerable novelist and critic George Eliot and her partner George Lewes—Clifford and his literary wife, Lucy, often attended their famous Sunday afternoon gatherings of writers, scientists, philosophers, and interested friends. Like Lewes, Eliot took an interest in science and maths. She had studied geometry in the 1850s at an adult class “for the ladies”—in 1882 Tait would grumble to Cayley that he was “virtually forced” to give a course of lectures for “ladies,” but presumably the problem was with his workload, not the women. Now, in the 1870s, Eliot was reading up on the new, non-Euclidean geometry that we’ll see more of in chapter 10, and in which Clifford also made his mark.24

至于刘易斯,他是一位哲学家、评论家、传记作家和出色的讲故事者。1873 年,他完成了一本新书《生命和心灵的问题》。这是对意识可能存在的生物学基础的法医、经验主义探索,是当时“新心理学”的一部分,它为长期存在的“身心问题”——意识(心灵)和大脑(身体)之间有争议的关系——带来了更现代的科学方法。正如我们所见,19 世纪是数学和物理学极具创造力的世纪,但正如麦克斯韦在 1856 年告诉他的学生一样,物理学是最简单的科学,它没有告诉我们任何有关情感、灵性或我们生活中其他“更高级”方面的事情。刘易斯希望通过将物理学中发展起来的经验主义和科学严谨性应用于心理学的生物学研究来探索这些“更高级的方面”。

As for Lewes, he was a philosopher, critic, biographer, and a marvelous raconteur. In 1873 he was finishing a new book, Problems of Life and Mind. It was a forensic, empiricist exploration of the possible biological basis of consciousness, part of “the new psychology” of the time, which was bringing a more modern scientific approach to long-standing debates on the “mind-body problem”—the contested relationship between consciousness (mind) and the brain (body). The nineteenth century had been an extraordinarily creative one for mathematics and physics, as we’ve seen, but as Maxwell had told his students back in 1856, physics is the simplest science, and it tells us nothing about emotions, spirituality, or other “higher” aspects of our lives. Lewes wanted to explore these “higher aspects” by applying to the biological study of psychology the kind of empiricism and scientific rigour developed in physics.

然而,与物理学的惊人进步相比,生物学的进步却存在问题。将化合物或电路分解成其组成部分是一回事,但在生物学中,这种还原主义的方法会带来严重的性别歧视和种族主义后果。例如,人们发现女性的脑容量比男性小,这立即被认为证明了文化期望,即女性不适合从事繁重的脑力劳动。萨默维尔和杰曼在这种假设下遭受了巨大损失,尽管她们最终克服困难取得了成功,但大多数女性甚至都没有尝试过。同样的伪科学方法助长了种族主义。就连 19 世纪生物学领域最耀眼的明星——自然选择进化论——也被用来“证明”穷人、女性和非欧洲人低人一等的文化观念。查尔斯·达尔文是艾略特和刘易斯的朋友;赫伯特·斯宾塞也是如此,他的名字与最糟糕的“社会达尔文主义”联系在一起,但他可能更像是一个“自由”功利主义者。他也是一位哲学家和先驱社会学家,尽管麦克斯韦认为他把社会进化的类比推得太远了,他和泰特认为斯宾塞的形而上学表明他对物理学的理解不够充分。25

Yet by contrast with physics’ dramatic progress, advances in biology had been problematic. It is one thing to break a chemical compound or an electric circuit into its constituent parts, but in biology such a reductionist approach had dire sexist and racist consequences. For instance, the discovery that women had smaller brains than men was immediately assumed to prove the cultural expectation that they were unfit for strenuous intellectual work. Somerville and Germain suffered greatly under this assumption, and although they eventually succeeded against the odds, most women did not even try. The same pseudoscientific approach helped entrench racism. Even that most brilliant star in biology’s nineteenth-century firmament— the theory of evolution by natural selection—was being co-opted into “proving” cultural tropes about the inferiority of the poor, of women, and of non-Europeans. Charles Darwin was a friend of Eliot and Lewes; so was Herbert Spencer, whose name became linked with the worst kind of “social Darwinism” but who was perhaps more a “liberal” utilitarian. He was also a philosopher and a pioneering sociologist, although Maxwell thought he’d pushed the social evolution analogy too far, and he and Tait thought Spencer’s metaphysics showed an inadequate grasp of physics.25

然而,刘易斯掌握了最新的物理学,并在《生命和心灵问题》中恰当地引用了泰特和麦克斯韦的科学著作。他是一位优秀的学者——今天,他的作品仍被视为与身心之争相关的内容——他请泰特检查证明。当泰特终于抽出时间阅读时,他将自己的评论发给了刘易斯(和艾略特)的出版商、爱丁堡的约翰·布莱克伍德,并补充道:“我发现这本书读起来相当僵硬,比两倍的分析 [数学] 公式还要糟糕:但它确实非常有趣,会彻底激怒所谓的形而上学家。”不幸的是,布莱克伍德认为这本书读起来僵硬了,所以刘易斯不得不寻找另一家出版商。26

Lewes, however, was on top of the latest physics and referred pertinently to Tait’s and Maxwell’s scientific writings in his Problems of Life and Mind. Like the good scholar he was—and today his work is still seen as relevant to the mind-body debate—he asked Tait to check the proofs. When Tait finally managed to find the time to read them, he sent his comments to Lewes’s (and Eliot’s) publisher, Edinburgh-based John Blackwood, adding, “I found it pretty stiff reading, far worse than double its amount of analytical [mathematical] formulae: but it is very interesting indeed, and will thoroughly rile the so-called Metaphysicians.” Unfortunately, Blackwood decided it was too stiff a read, and so Lewes had to look around for another publisher.26

泰特认识布莱克伍德,不仅仅是因为他是刘易斯的裁判,两人有时还一起打高尔夫球。在一次打高尔夫球之后,泰特向布莱克伍德展示了麦克斯韦写的一首诗,这首诗总结了最近在贝尔法斯特举行的英国协会会议上主席的致辞——麦克斯韦和克利福德出席了会议,但泰特没有出席。会议的主要议题是进化论和原子的科学地位——这是 1874 年,比发现电子早 25 年,比原子核早近 40 年。麦克斯韦支持原子论,但他不是唯物主义者,他用他邪恶的机智嘲笑了“我们愚蠢的贵族”之间无休止的“英国蠢货”讨论。你可以在开场白中看到它的主旨:

It wasn’t just as Lewes’s referee that Tait knew Blackwood, for the two men sometimes played golf together. After one such round, Tait showed Blackwood a poem Maxwell had written him, summing up the President’s Address at a recent British Association meeting in Belfast—Maxwell and Clifford had attended but not Tait. The big topics of the meeting were evolution, and the scientific status of atoms—this was in 1874, twenty-five years before the electron was discovered and almost four decades before the nucleus of the atom. Maxwell supported the idea of atoms, but he was not a materialist, and with his wicked wit he made great fun of endless “British Ass” discussions among “our witless nobs.” You can see the tenor of it in the opening lines:

在科学发展的初期,管理事物的牧师们

善于使用锤子和凿子,以人的形象造神…… 27

In the very beginning of science, the parsons, who managed things then,

Being handy with hammer and chisel, made gods in the likeness of men….27

最终,我们得到了商业,诗中继续写道,还有尘世的力量,最后原子“取代了魔鬼和神”,成为“质量和思想的单位”——“无结构的细菌”,使我们能够“继承野兽、鱼类和蠕虫的思想”。这首诗既涉及科学,也涉及宗教——或者更确切地说,涉及目前双方更极端的支持者之间的争论——布莱克伍德很喜欢它。他甚至提出把它发表在他著名的《布莱克伍德杂志》上,该杂志早已出版了刘易斯的作品,也让乔治·艾略特成为了小说作家。事实上,布莱克伍德杂志最近出版了她的杰作,米德尔马契——当时许多小说都是以连载形式出现在杂志上的——因此麦克斯韦的匿名诗也名声显赫。他对这个邀请感到吃惊——他告诉泰特,英国驴子的私人“突发事件”不应该“被随意定型”;但最终这个想法太有趣了,他的“更好的一半”同意了。28

Eventually we got commerce, the poem continues, and earthly power, and finally atoms “supplanted both demons and gods,” and became “the unit of mass and of thought”—“structureless germs” that enabled us to “inherit the thoughts of beasts, fishes and worms.” The poem has a go at both science and religion—or rather, at the current debate between the more extreme proponents of either side—and Blackwood loved it. So much so that he offered to publish it in his famous Blackwood’s Magazine, which had long published Lewes, and had also given George Eliot her start as a fiction writer. In fact, Blackwood’s had recently published her masterpiece, Middlemarch—many novels at the time first appeared in serial form in magazines—so Maxwell’s anonymous poem was in illustrious company. He’d been taken aback by the invitation—he told Tait that the private “paroxysms” of British Asses should not be “promiscuously stereotyped”; but in the end the idea was such a hoot that his “better ½” agreed to it.28

尽管他的诗讽刺了牧师和唯物主义者,但麦克斯韦已经为进化的讨论做出了重要贡献,例如,他提出分子的有限尺寸将限制可以传递的遗传信息量。(达尔文最初认为遗传是绝对的,一切都是作为一个整体传递下来的——因此麦克斯韦嘲笑鱼和蠕虫的思想。)尽管他拒绝了大部分正式宗教,但麦克斯韦相信上帝创造了一切物质——而克利福德是一名无神论者,他认为麦克斯韦太快限制了原子的进化作用。29

Although his poem lampooned both parsons and materialists, Maxwell had already made important contributions to the discussion on evolution, suggesting, for example, that the finite size of molecules would limit the amount of genetic information that could be passed down. (Darwin had originally believed that heredity was absolute, that everything was passed down as a whole—hence Maxwell’s gibe about the thoughts of fishes and worms.) Still, although he’d rejected much of formal religion, Maxwell believed that God created all matter—and Clifford, who was an atheist, thought Maxwell had been too quick to limit the evolutionary role of atoms.29

1878 年,克利福德发表了关于向量的主要著作,同年还出版了《讲座与论文集》,其中有关于物质、身心问题和伦理学的科学方法的文章。麦克斯韦被要求对这本书进行审阅,但他发现这是一项棘手的任务——一段时间以来,他一直隐约感到身体不适,这让他的工作比平时更加​​困难。他告诉坎贝尔,克利福德的《讲座》中有很多东西需要“严厉批评”,但他想温和地表达自己的批评,因为“克利福德是个很好的人。” 30

In 1878, the year he published his major work on vectors, Clifford also published Lectures and Essays, with articles about matter, the mind-body problem, and a scientific approach to ethics. Maxwell was asked to review it, but he found it a delicate task—made more difficult than usual by the vague ill health that had been troubling him for a while now. He told Campbell there were many things in Clifford’s Lectures that needed “trouncing,” but that he wanted to express his criticisms gently, for “Clifford was such a nice fellow.”30

所有这些争论——来来回回、机智的讽刺和谨慎的批评——都凸显了当今科学界的巨大动乱,以及所有领军人物的广泛参与,无论他们的主要领域是什么。然而,在载体问题上,这种动乱即将爆发成激烈的辩论,就像进化论等更明显有争议的话题一样。不幸的是,麦克斯韦和克利福德将无法发挥他们的作用。

All this debate—the back and forth, the witty barbs and careful criticisms—highlights the great scientific ferment of the times, and the broad participation by all the leading lights, no matter their primary field. On the subject of vectors, however, this ferment is about to erupt into debates as heated as any of those on more obviously controversial topics such as evolution. Tragically, Maxwell and Clifford will not get to play their parts.

再见,麦克斯韦和克利福德

FAREWELL MAXWELL AND CLIFFORD

当麦克斯韦尔开始为克利福德撰写详细检查报告时,他的大脑完全无法工作——最后他意识到自己的病情很严重。医生宣布他得了腹部癌症,并说他只能活一个月了。

When Maxwell came to write his careful review of Clifford, his mind simply would not work—and at last he knew his illness was serious. The doctor declared it abdominal cancer and gave his patient only a month to live.

1879 年末,麦克斯韦去世,享年 48 岁。认识他的人都为麦克斯韦如此早逝而悲痛不已,他如此聪明、如此温和。正如坎贝尔所说,他的“伟大朴素”感动了所有人,人们纷纷悼念他。泰特为他写的讣告主要是一份朴素的传记摘要,尽管你可以看到他对童年好友的喜爱和钦佩。但正如泰特自己的传记作者兼前学生卡吉尔·诺特后来所说,“泰特对麦克斯韦的天才有着无限的钦佩,对他有着深厚的爱,对他的古怪和幽默有着敏锐的欣赏。” 31

Maxwell died at the age of forty-eight, in late 1879. Everyone who knew him was devastated at the loss, so early, of so bright and gentle a spirit. His “grand simplicity,” as Campbell put it, touched everyone, and the tributes poured in. Tait’s obituary for him is primarily a sober biographical summary, although you can see glimpses of the affection and admiration he had for his childhood friend. But as Tait’s own biographer and former student Cargill Knott would later put it, “Tait had an unstinted admiration for the genius of Maxwell, a deep love for the man, and a keen appreciation of his oddities and humour.”31

一场感人的葬礼在麦克斯韦尔的三一学院举行。随后,他被送回了格伦莱尔的家中,至今仍安息在附近一座孤零零的教堂墓地里。

A moving funeral service was held at Maxwell’s college, Trinity. Then he was taken home to Glenlair, and he still lies in the lonely churchyard nearby.

• • •

• • •

克利福德也走了,而且死得太早了,1879 年年初,他因肺结核去世,年仅 33 岁。多年后,剧作家萧伯纳告诉露西·克利福德,她的丈夫比任何人都聪明,除了阿尔伯特·爱因斯坦——还冷冷地补充道:“甚至比我还要聪明!” 32

Clifford was gone, too, and far too soon, for he was only thirty-three when he died of tuberculosis earlier in 1879. Years later, the playwright George Bernard Shaw told Lucy Clifford that her husband had been smarter than anyone except Albert Einstein—adding drily, “cleverer even than ME!”32

坎贝尔为他的童年好友麦克斯韦所写的温情传记,最后一页致敬了麦克斯韦和克利福德,这篇传记刊登在了《蓓尔美尔报》上。在下一章中,我们将会介绍一些继续从事这项工作的年轻人,以此来结束这一章,这是一种恰当的方式。文章写道:“麦克斯韦的名字得到了所有在这类问题上有权发言的人的一致同意,并且处于最重要的地位。”他拥有罕见而又具有影响力的天才,他“与上一代的法拉第……以及同为数学家的克利福德……在很多方面都与麦克斯韦相提并论。”如果他们活得更久,谁知道他们还能取得什么样的成就。

In Campbell’s tender biography of his childhood friend Maxwell, the last page includes a tribute to both him and Clifford, which was published in the Pall Mall Gazette. It is a fitting way to end this chapter, before we meet in the next some of the young men who carried on their work. “Maxwell’s name stands,” the article said, “by the unanimous consent of all who have any voice in such matters, in the very foremost rank.” He had a rare yet influential genius, a quality he shared “with Faraday in an earlier generation … and with Clifford, a fellow mathematician … whose intellect was in more ways than one akin to Maxwell’s own.” Had they lived longer, who knows what else they might have accomplished.

(8)向量分析的最终成果——以及四元数的“战争”

(8) VECTOR ANALYSIS AT LAST—AND A “WAR” OVER QUATERNIONS

麦克斯韦注定无法看到那些受到他的《电磁论》启发的年轻人所取得的成就。这一小群追随者被称为“麦克斯韦派”,其中包括乔治·菲茨杰拉德,他从麦克斯韦方程中推导出一种检测电磁波的理论方法;他的朋友和合作者奥利弗·洛奇;以及海因里希·赫兹。洛奇和赫兹首次在实验室中独立成功产生这些波——惊人地证实了麦克斯韦的理论。赫兹在 1888 年发表了他的研究成果,可怜的洛奇——他正准备在即将召开的英国协会会议上展示自己的研究成果——却发现自己被抢先了。1

Maxwell wasn’t fated to see the achievements of the young men who were inspired by his Treatise on Electricity and Magnetism. This small band of devotees has been dubbed “the Maxwellians,” and it includes George FitzGerald, who deduced from Maxwell’s equations a theoretical way of detecting electromagnetic waves; his friend and collaborator Oliver Lodge; and Heinrich Hertz. Lodge and Hertz independently succeeded in generating these waves in the lab, for the very first time—spectacularly confirming Maxwell’s theory in the process. Hertz presented his results in 1888, and poor Lodge—who had been preparing to present his own findings at the upcoming British Association meeting—discovered he’d been scooped.1

到 1901 年,技术已经得到极大改进,古列尔莫·马可尼和他的团队能够发送第一个长距离电报信号,信号飞越大西洋,而不是像威廉·汤姆森那样在海底传输——无线时代就此诞生。事实上,马可尼的信号是由麦克斯韦的前学生安布罗斯·弗莱明设计的发射器发送的。

By 1901, the technology had improved so much that Guglielmo Marconi and his team were able to send the first long-distance telegraphic signals, soaring across the Atlantic Ocean rather than under it as William Thomson’s had done—and the wireless era was born. In fact, Marconi’s signal was sent from a transmitter designed by Maxwell’s former student Ambrose Fleming.

然而,对于我们的故事来说,麦克斯韦的关键人物是一位古怪的电报员,奥利弗·亥维赛德——因为他扩展了麦克斯韦的全矢量方法,并将其转变为现代矢量分析。

For our story, though, the key Maxwellian is an eccentric telegrapher, Oliver Heaviside—for he is the one who extended Maxwell’s whole-vector approach and turned it into modern vector analysis.

到 19 世纪 80 年代,已铺设了数千英里的电缆,电报已成为一项引人注目的新技术,类似于 21 世纪初的信息技术。与最近 IT 课程的流行一样,汤姆森报告了格拉斯哥大学实验课上想要成为电报工程师的年轻学生的“流行病”。然而,年轻的赫维赛德从未上过大学。19 世纪新技术爆炸式增长的负面影响是工匠的工作减少,包括赫维赛德的父亲,他是木刻师,他的技能现在必须与新的摄影技术竞争以复制图像。他的母亲试图通过经营一所小学校来维持生计,但赫维赛德在贫困的边缘长大——并且厌恶伦敦的贫民窟、那里的骗子和醉鬼酒吧。2

By the 1880s, many thousands of miles of cables had been laid, and telegraphy had become a glamorous new technology, akin to information technology in the early 2000s. And much like the recent popularity of IT courses, Thomson reported on the “epidemic” of young students, in his lab classes at the University of Glasgow, who wanted to train as telegraphic engineers. Young Heaviside, however, never went near a university. The downside of the nineteenth-century explosion of new technology was the erosion of work for craftspeople, including Heaviside’s wood-engraver father, whose skills now had to vie with new photographic techniques for reproducing images. His mother tried to make ends meet by running a small school, but Heaviside grew up on the edge of poverty—and with a loathing for the mean streets of London with their cheating tradespeople and their boozy pubs.2

16 岁时,他跟随两个哥哥开始为叔叔查尔斯·惠斯通 (Charles Wheatstone) 做电报员。惠斯通在 19 世纪 30 年代曾参与设计了最早的电报系统之一,并与海维赛德的姨妈结婚。大约一年后,即 1868 年,海维赛德找到了一份工作,负责研究新的英丹电报电缆,测试线路上的信号速度——他对电及其测量的技术性非常着迷,他研究了所有能找到的相关信息。他发现汤姆森早期关于电报信号传输的理论工作特别有帮助,所以他已经不再是普通的电报员了。

Following two of his older brothers, he started work at sixteen as a telegrapher for his uncle, Charles Wheatstone, who had codesigned one of the earliest electrical telegraph systems back in the 1830s, and who had married Heaviside’s maternal aunt. A year or so later, in 1868, Heaviside landed a job working on the new Anglo-Danish telegraph cable, testing the signal speed along the line—and he became so fascinated by the technicalities of electricity and its measurement that he studied all he could find about it. He found Thomson’s early theoretical work on telegraphic signal transmission particularly helpful, so already he was no ordinary telegrapher.

他的深思熟虑终于得到了回报,他发表了一篇论文,这篇论文成为他获得学术尊重的敲门砖。这篇论文是关于如何使用惠斯通电桥——一种用于测量电阻的装置,以他叔叔的名字命名,他叔叔改进并推广了塞缪尔·克里斯蒂的原始设计。你需要能够准确地测量电阻,这样你才能调节信号,确保电缆不会过热。汤姆森对赫维赛德的论文印象深刻,亲自向他表示祝贺。

His deep thinking paid off when he published a paper that proved to be his ticket to academic respectability. It was on the best way to use a Wheatstone bridge—a device for measuring electrical resistance, named for his uncle, who had improved and popularised the original design by Samuel Christie. You needed to be able to measure resistance accurately so you could regulate your signals and make sure your cables wouldn’t overheat. Thomson was so impressed with Heaviside’s paper that he personally congratulated him.

麦克斯韦也对此印象深刻。年轻的赫维赛德的人生开端很艰难——猩红热使他听力下降——但他信心十足,并给麦克斯韦寄了一份论文。1873 年 2 月,麦克斯韦即将完成《数学分析》的最后出版工作,但最后一刻,他添加了对该论文的引用以及其主要结果的简要概述。当年晚些时候,当海维赛德浏览《数学分析》时,他被震撼了——不仅仅是因为他自己的工作被提及。相反,他对麦克斯韦这本书的深度和广度感到敬畏:“我看到它很伟大、更伟大、最伟大”,他兴奋地说道,于是他开始掌握它。3

Maxwell was impressed, too. Young Heaviside had had a rough start in life—made worse by scarlet fever that had left him partially deaf—but he had oodles of confidence and had sent Maxwell a copy of his paper. It had been published in February 1873, when Maxwell was in the final stages of seeing his Treatise into print, but he added in at the last minute a reference to the paper along with a brief outline of its key result. When Heaviside browsed through the Treatise later that year, he was blown away—and not just because his own work scored a mention. Rather, he was awed by the marvelous depth and breadth of Maxwell’s book: “I saw that it was great, greater, and greatest” he enthused, and so he set out to master it.3

与此同时,他在行业中也做得很好。多年来,他积累了大量的电气知识——也得益于与兄弟亚瑟的合作——他的事业似乎蒸蒸日上。但突然间,在 1874 年,也就是他 24 岁时,他辞去了工作,回到父母身边生活。没有人知道原因。有人认为,他性格暴躁,可能让他的打工生活变得艰难。但他似乎也爱上了电气理论——以及“上天派来的”麦克斯韦——并想有时间学习和写论文。4

At the same time, he was doing well in his trade. Over the years he’d gained a great deal of electrical knowledge—helped also by collaborations with his brother Arthur—and his career seemed to be on the up and up. But suddenly, in 1874 when he was twenty-four, he quit his job and returned to live with his parents. No one knows quite why. It’s been suggested that his prickly personality may have made life as an employee difficult. But it also seems that he’d fallen in love with electrical theory—and with the “heaven sent” Maxwell—and wanted the time to study and write papers.4

无论出于什么原因,终身单身的海维赛德过着越来越孤立的生活,致力于撰写越来越复杂的电气和电报理论文章,尤其是为每周贸易杂志《电工》撰写文章。他的第一篇关于麦克斯韦理论的论文《论电流的能量》发表于 1883 年。

Whatever the reason, Heaviside, a lifelong bachelor, lived an increasingly isolated life devoted to writing increasingly sophisticated articles on electrical and telegraphic theory, especially for the weekly trade magazine The Electrician. His first paper on Maxwell’s theory, “On the Energy of Electric Currents,” appeared in 1883.

他还撰写了数学方面的论文,包括他在《麦克斯韦论文》中首次发现的矢量分析。正是海维赛德引入了清晰、粗体的矢量字体,并因认为四元数在实际应用中毫无用处而与泰特产生了矛盾。他在《电磁理论》一书中开玩笑地写道:

He also wrote on mathematics—including vector analysis, which he first discovered in Maxwell’s Treatise. It is Heaviside who introduced the clear, bold typeface for vectors, and who got on the wrong side of Tait by dismissing quaternions as useless for practical purposes. He wrote facetiously, in his book Electromagnetic Theory,

我认为,一位美国女学生将“四元数”定义为“一种古老的宗教仪式”。然而,这完全是错误的。古人——不像泰特教授——不知道四元数,也不崇拜四元数。5

“Quaternion” was, I think, defined by an American schoolgirl to be “an ancient religious ceremony.” This was, however, a complete mistake. The ancients—unlike Professor Tait—knew not, and did not worship, Quaternions.5

赫维赛德经常很搞笑,即使是在技术文章的中间!他也常常很有智慧——比如当他在讲述麦克斯韦的中途停下来时电磁创新,瞄准那些以“挑剔和不接受的态度”批评他们的科学家。他指的是有人抱怨麦克斯韦的最终理论纯粹是数学的,没有电磁如何从一个地方传播到另一个地方的物理模型。

Heaviside is often hilarious, even in the middle of a technical passage! He’s also often wise—as when he pauses midway in an account of Maxwell’s electromagnetic innovations, taking aim at scientists who criticised them in “a carping and unreceptive spirit.” He was referring to the complaint that Maxwell presented his final theory purely mathematically, with no physical model for how electromagnetism was supposed to get from one place to another.

和麦克斯韦一样,海维赛德关注的是向量及其标量和向量积,而不是四元数——但他更进一步,放弃了所有将向量作为四元数虚部的说法。“我从来不明白,”他说。“读者应该彻底放弃我们关心虚部的想法1在矢量分析中。” 6毕竟,正如我之前指出的那样,无论使用虚基还是实基,矢量的分量都是相同的。因此,多亏了海维赛德,汉密尔顿的虚数i, j, k才变成了今天的i, j, k

Like Maxwell, Heaviside focussed on vectors and their scalar and vector products, rather than on quaternions—but he went further by discarding all reference to vectors as the imaginary part of a quaternion. “I never understood it,” he said. “The reader should thoroughly divest his mind of any idea that we are concerned with the imaginary 1 in vector analysis.”6 After all, as I pointed out earlier, the components of a vector are the same whether you use an imaginary or real basis. So, it is thanks to Heaviside that Hamilton’s imaginary i, j, k became today’s i, j, k.

通过使i、jk成为平方为 1 的实单位向量,而不是虚数平方得到的 −1,还可以去掉我在上一章中展示的哈密顿标量积中的负号。这个不必要的符号也让麦克斯韦烦恼不已,但海维赛德将它抛在一边。“向量的平方怎么会是负数?”他问道——并幽默地补充道,“而哈密顿对此非常肯定。”他这种奇妙的无忧无虑也许反映了他局外人的身份:粗略地说,物理学家不认真对待他,因为他没有上过大学,而实践者不认真对待他,因为他们认为理论是给小白看的。7

By making i, j, and k real unit vectors whose square is 1, rather than the −1 that comes from squaring imaginary numbers, you also get rid of the minus sign in Hamilton’s scalar product that I showed in the previous chapter. This unnecessary sign had annoyed Maxwell no end, too, but it was Heaviside who tossed it aside. “How could the square of a vector be negative?” he asked—adding drolly, “And Hamilton was so positive about it.” His wonderful insouciance is perhaps a reflection of his outsider status: speaking loosely, the physicists didn’t take him seriously because he hadn’t been to university, and the practitioners didn’t take him seriously because they thought theorising was for nobs.7

麦克斯韦学派是这一规则的一个显著例外,汤姆森也是如此。1888 年,在巴斯举行的英国物理学会会议上,失望的洛奇在赫兹先前在德国宣布的阴影下展示了他发现的电磁波,会议还讨论了如何解释麦克斯韦的理论。洛奇、菲茨杰拉德和汤姆森都知道并钦佩赫维赛德在这方面的工作,他们大力推广,以至于会议的官方记录指出,“每个人都对赫维赛德先生的缺席表示遗憾。”赫维赛德本人是个孤僻的人,他从来没有打算参加任何科学聚会。但最后,物理学家们注意到了他。8

The Maxwellians were notable exceptions to this rule—and so was Thomson. At the 1888 meeting of the British Association in Bath—where the disappointed Lodge presented his discovery of electromagnetic waves in the shadow of Hertz’s prior announcement in Germany—there was also discussion about how to interpret Maxwell’s theory. Lodge, FitzGerald, and Thomson all knew and admired Heaviside’s work on this, and they promoted it so well that the official write-up of the meeting noted that “everyone expressed regret at the absence of Mr. Heaviside.” Heaviside himself, loner that he was, had no intention of ever attending a scientific gathering. But at last, the physicists were taking notice of him.8

赫维赛德改变了麦克斯韦方程组的形式

HEAVISIDE FAMOUSLY CHANGES THE FORM OF MAXWELL’S EQUATIONS

巴斯的讨论涉及麦克斯韦数学量和方程的物理解释。我们已经看到汤姆森是那些认为麦克斯韦理论数学上令人印象深刻但没有电磁波传播物理模型的人之一——但海维赛德一直在问一个不同的问题。作为一名电报员,他感兴趣的是如何发送信号——他将其视为电能——而不失真。麦克斯韦曾写过关于电场和磁场能量的文章,海维赛德将其从𝔈和𝔅改为现代的EB;但麦克斯韦显然更喜欢“势”的数学优雅,而不是更直接地强调物理场。

The discussion in Bath concerned the physical interpretation of Maxwell’s mathematical quantities and equations. We’ve seen that Thomson was one of those who’d dismissed Maxwell’s theory as having impressive maths but no physical model for how electromagnetic waves propagated—but Heaviside had been asking a different question. As a telegrapher, he was interested in the problem of sending signals—which he saw as electrical energy—without distortion. Maxwell had written on the energy of the electric and magnetic fields, which Heaviside turned from 𝔈 and 𝔅 to the modern E and B; but Maxwell apparently preferred the mathematical elegance of the “potentials” to a more direct emphasis on the physical fields.

具体来说,正如我在第 7 章中所展示的,他选择用矢量势来表示磁场:

In particular, and as I showed in chapter 7, he chose to represent the magnetic field by a vector potential:

𝔅 = V . ∇𝔘

(现代符号为B = ∇ × A ),

𝔅 = V. ∇𝔘

(B = ∇ × A in modern notation),

其中他的 𝔘 和现代的A代表电磁场的矢量势。这是一种巧妙的数学方法,因为这样你就可以使用矢量微积分恒等式(例如旋度的散度始终为零)来推导出进一步的方程式——在本例中,B的散度为零:∇ ∙ B = 0,或者如海维赛德所写,div B = 0。(像克利福德一样,他将麦克斯韦的“收敛”改为现代的“散度”。他将 ∇ × A写为 curl A。)

where his 𝔘 and the modern A represent the vector potential of the electromagnetic field. It’s a neat way to do it mathematically, because then you can use vector calculus identities, such as the divergence of a curl is always zero, to deduce further equations—in this case, that the divergence of B is zero: ∇ ∙ B = 0, or as Heaviside wrote it, div B = 0. (Like Clifford, he made the change from Maxwell’s “convergence” to the modern “divergence.” And he wrote ∇ × A as curl A.)

类似地,麦克斯韦使用牛顿点符号作为时间导数的简写,将电场表示为

Similarly—and using Newton’s dot notation as shorthand for the time derivative—Maxwell had represented the electric field as

𝔈 = V。 𝔊𝔅 −𝔘− ∇Ψ,

𝔈 = V. 𝔊𝔅 −𝔘− ∇Ψ,

或者用现代符号来表达,

or in modern symbols,

=×d一个dφ

E=v×BdAdtϕ,

其中 Ψ(或 φ)是电势或电压,v是任何带电粒子的速度,这些粒子的运动对磁场有贡献(如根据 Øersted 的说法,这反过来又对电场有贡献(根据法拉第的说法)。如果电场仅由变化的磁场引起,则v = 0。

where Ψ (or ϕ) is the electric potential or voltage, and v is the velocity of any charged particle whose motion is contributing to a magnetic field (as per Øersted) that in turn contributes to the electric field (as per Faraday). If the electric field is due to a changing magnetic field alone, then v = 0.

海维赛德承认,有时数学势方法对计算很有用,但他想直接研究电磁能的物理原理。当他在方程中加入势时,他引入了简写“pot”——就像他用“div”和“curl”表示散度和旋度算子一样——并调皮地补充说,在这种情况下,“pot……与 kettle 的关系并不比三角函数与不言而喻的那个关系更密切。似乎没有必要这么说,但也不能太过挑剔。”确实如此!但​​在整理与能量传输有关的麦克斯韦方程时,海维赛德“扼杀”了势,正如他告诉菲茨杰拉德的那样——而且理由充分。在《电子与分子动力学分析》中(以及在他 1865 年的原始论文中),麦克斯韦以矢势 𝔘(或A )的形式推导出光的波动方程(或任何电磁信号的波动方程),而海维赛德则证明,在分析电磁能的传输时,这是不符合物理规律的。9

Heaviside allowed that sometimes the mathematical potential approach could be useful for calculations, but he wanted to get straight down to the physics of electromagnetic energy. When he did include a potential in an equation, he introduced the shorthand “pot”—just as he used “div” and “curl” for the divergence and curl operators—adding mischievously that in this case, “pot … has no more to do with kettle than the trigonometrical sin has to do with the unmentionable one. It seems unnecessary to say so, but one cannot be too particular.” Quite! But in sorting out Maxwell’s equations in relation to the transmission of energy, Heaviside “murdered” the potentials, as he told FitzGerald—and with good reason. In the Treatise (and in his original 1865 paper), Maxwell had derived his wave equation for light—or for any electromagnetic signal—in terms of the vector potential 𝔘 (or A), and Heaviside had shown that this was unphysical when it came to analysing the transmission of electromagnetic energy.9

因此,现在海维赛德的主要目标是重塑麦克斯韦方程,使物理场EB成为主要事件——就像今天一样(至少一般而言)。10不知道的是,在 1868 年的一篇论文中,麦克斯韦已经直接根据场导出了波动方程。它非常简单,可惜他没有进一步发展这种方法。相反,正是海维赛德,那个尖酸刻薄的自学成才的孤独者,将这种方法带入了主流。11

So now Heaviside’s main goal was to recast Maxwell’s equations so that the physical fields, E and B, would be the main event—just as they are today (generally speaking, at least).10 What he didn’t know was that in an 1868 paper, Maxwell had derived a wave equation directly in terms of the field. It’s beautifully simple, and it’s a pity he didn’t take this approach further. Instead, it is Heaviside, that acerbic self-taught loner, who brought this approach into the mainstream.11

事实上,你可能在流行书籍和博客中读到过,在麦克斯韦的论文中,有 20 个分量形式的“麦克斯韦方程”,而正是海维赛德将这些方程简化为四个美丽的全矢量方程,这些方程如今在教科书(和 T 恤上)中很有名:

In fact, you may have read in popular books and blogs that in Maxwell’s papers, there were twenty component-form “Maxwell’s equations,” and that it was Heaviside who reduced these to the four beautiful whole-vector equations that are famous today in textbooks (and on T-shirts):

  =4πρ  =0×=×=+4πJ

 E =4πρ B =0×E=Bt×B=Et+4πJ

(其中 ρ 是电荷密度,J是电流密度)。12然而,海维赛德的这种说法并不完全是事实。首先,让我们重新看看这些著名方程的含义。前两个方程是散度方程,与流过电荷或磁体周围封闭曲面的磁通量有关,如图6.17.1所示。这两个方程是高斯-库仑定律的矢量形式。第三和第四个方程与法拉第定律和安培定律有关,涉及场的旋度。正如我在图 7.2中所示,旋度运算与旋转有关。因此, ∇ × E方程表示实验观察到的事实,即变化的磁场会感应出绕环流动的电流,而 ∇ × B方程则表示感应磁场会“旋转”或绕着载流导线转圈。 (如果你把铁屑撒在导线穿过的木板上,你会看到铁屑排列成圆圈。)这两个方程右边的导数表明,电场和磁场如果要互相感应,就必须随时间变化。

(where ρ is the charge density and J is the current density).12 This claim on behalf of Heaviside is not quite the true picture, however. First, though, let’s look again at what these famous equations mean. The first two are divergence equations, which relate to the flux flowing through a closed surface around the charge or magnet, as illustrated in figures 6.1 and 7.1. These two equations are vector forms of the Gauss-Coulomb laws. The third and fourth equations relate to Faraday’s and Ampère’s laws, and they involve the curl of the fields. As I showed in figure 7.2, the curl operation relates to rotations. So, the ∇ × E equation encodes the experimentally observed fact that a changing magnetic field induces a current that flows around a loop, while the ∇ × B equation shows that an induced magnetic field “rotates” or circles around a current-carrying wire. (If you scatter iron filings on a board through which the wire passes, you’ll see the filings line up in circles.) The derivatives on the right-hand sides of these two equations show that the electric and magnetic fields must be changing in time if they are to induce each other.

这是一组极其优雅且强大的方程式。

It’s an extraordinarily elegant and powerful set of equations.

从麦克斯韦到这四个美丽方程的故事比之前长期被忽视的海维赛德的流行传说要复杂一些——这些事情通常都是如此。除了海维赛德写的是“div”和“curl”而不是点和叉之外,据我所知,他从来没有把这四个方程写在一起:就像麦克斯韦的主要场方程中穿插着用于各种应用或定义各种术语的方程一样,海维赛德也是如此。原创研究也是如此,因为你在进行过程中会提出你的定义、理由和应用。他确实把两个旋度方程写在一起了——他称之为“双重方程”——有时他也会把两个散度方程写在一起,但即便如此,这两对方程也散布在几十个其他方程之中。更重要的是,海维赛德并没有以相当现代的形式写出这四个方程:他对它们进行了调整,使它们更加对称,正如你在尾注中看到的那样。13

The story of how we went from Maxwell to these four beautiful equations is a little more complicated—as these things usually are—than the popular legend that has grown up around the previously long-neglected Heaviside. Aside from the fact that Heaviside wrote “div” and “curl” instead of using the dot and cross, as far as I’m aware he never wrote these four equations all together: just as Maxwell’s main field equations were interspersed with equations for various applications or for defining various terms, so were Heaviside’s. That’s how it goes with original research, for you are presenting your definitions, justifications, and applications as you go. He did write his version of the two curl equations together—he called them “duplex equations”—and he sometimes wrote the two divergence equations together, but even then, the two pairs of equations were interspersed among dozens of other equations. What’s more, Heaviside didn’t write these four equations in quite the modern form: he tweaked them to make them more symmetrical, as you can see in the endnote.13

然而,比这些细节更重要的是,麦克斯韦在他的《数学论》中,实际上把他的二十个分量方程简化为五个全矢量方程。撇开符号不谈,这两个方程对于EB的势,我之前已经展示过:第一个等价于 ∇ ∙ B = 0,正如我们所见,第二个等价于×=(我将在下一个尾注中展示)。然后是我在第 7 章中给出的 ∇ ∙ E方程,加上B旋度的方程,他这样写:

Much more important than these details, however, is the fact that in his Treatise Maxwell had, in fact, reduced his twenty component equations to just five whole-vector ones. Notation aside, these are the two equations for E and B in terms of potentials, which I showed earlier: the first is equivalent to ∇ ∙ B = 0, as we saw, and the second is equivalent to ×E=Bt (as I’ll show in the next endnote). Then there’s the ∇ ∙ E equation I gave in chapter 7, plus an equation for the curl of B, which he wrote like this:

4πℭ = V.∇ℌ

4πℭ = V. ∇ℌ.

对于纯磁场,他定义 ℌ 与 𝔅 成正比,定义 ℭ 为传导电流加上电位移 𝔇 的导数(正如我们在上一章中简要介绍过的,它与 𝔈 成正比)——因此,这意味着,用现代符号和适当的单位表示,他的旋度方程确实是,

He defined ℌ to be proportional to 𝔅 for a purely magnetic field, and he defined ℭ to be the conduction current plus the derivative of the electric displacement 𝔇 (which is proportional to 𝔈 as we saw briefly in the previous chapter)—so this means, in modern notation with appropriate units, that his curl equation is, indeed,

4πJ+=×

4πJ+Et=×B

麦克斯韦还给出了一个附加方程,用于描述电场和磁场产生的机械力——正如海维赛德所说,这正是工程师在设计发电机和马达时所需要的。它本质上就是现在所谓的洛伦兹力定律——以我们在第 4 章中与电子自旋相关的洛伦兹命名。

Maxwell also gave an additional equation, for the mechanical force created by the electric and magnetic fields—and this, as Heaviside said, is what engineers need when designing dynamos and motors. It is essentially what is now called the Lorentz force law—named after the same Lorentz we met in chapter 4 in connection with electron spin.

因此,除了麦克斯韦 1868 年的论文之外,海维赛德所做的新奇之举是用EB而不是电位重写麦克斯韦方程(并将那些难以阅读的哥特式字母改为清晰的粗体字母)。这在那四个著名的方程中展现了电场和磁场之间的美丽对称性,这是海维赛德的妙招。在另一个独立的共同发现案例中,赫兹也做了类似的事情。(不幸的是,赫兹没有机会进一步推进他的工作。他于 1892 年因败血症去世,当时他只有 36 岁——所以他没有活着看到他的无线电波即将找到的非凡用途,就像麦克斯韦没有看到赫兹以惊人的方式证实了他对那些波的预测。)尽管如此,从电位到场矢量的这种变化很容易从麦克斯韦方程的版本中得出。我已经证明了麦克斯韦方程有 ∇ ∙ B、 ∇ ∙ E和 ∇ × B方程的全矢量等价形式;从麦克斯韦方程得到 ∇ × E方程也相对简单正如您在尾注中看到的那样。14就是为什么海维赛德从未声称这些是他的方程:它们都在那里,加上力方程,在麦克斯韦的《引力分析》中的五个全矢量方程中。15

So, the novel thing that Heaviside did—Maxwell’s 1868 paper aside— was to rewrite Maxwell’s equations in terms of E and B rather than the potentials (and to change those hard-to-read Gothic letters to clear boldface ones). This brings out, in those four famous equations, the beautiful symmetry between the electric and magnetic fields, and it was a masterstroke by Heaviside. In yet another case of independent codiscovery, Hertz had done something similar. (Tragically, Hertz never got the chance to take his work much further. He died of blood poisoning in 1892, when he was just thirty-six years old—so he never lived to see the extraordinary uses his radio waves were about to find, just as Maxwell never got to see the spectacular way in which Hertz had confirmed his prediction of those waves.) Still, this change from potentials to field vectors follows easily from Maxwell’s version of the equations. I’ve already shown that Maxwell had wholevector equivalents of the ∇ ∙ B, ∇ ∙ E, and ∇ × B equations; getting the ∇ × E equation from Maxwell’s equation for E is also relatively straightforward, as you can see in the endnote.14 That’s why Heaviside never claimed that these were his equations: they’re all there, plus the force equation, in those five whole-vector equations in Maxwell’s Treatise.15

海维赛德确实提出了对称性,并且他将重点放在EB上,以使方程更加实用——但他的“谋杀”势在当今的数学物理中仍然非常活跃。矢量势A可能不是物理的——EB才是可直接测量的——但正如海维赛德自己所说,它们通常使计算变得更简单。这种势的计算应用今天通常被称为“规范理论”:其思想是选择一个“规范”——一个用A和 ϕ 表示的方程——它可以简化计算而不改变EB的值。换句话说,电场和磁场在“规范变换”下保持不变——类似于无论你如何转动旋转的球看起来都是一样的。规范现在不仅用于简化电磁学中的计算,还用于量子理论和相对论。

Heaviside did bring out the symmetry, and he put the focus firmly on the E and B to make the equations more practical—but his “murdered” potentials are very much alive and well in mathematical physics today. The vector potential A may not be physical—it is E and B that are directly measurable— but as Heaviside himself said, they often enable calculations to be done more simply. This computational use of potentials is often known today as “gauge theory”: the idea is to choose a “gauge”—an equation in terms of A and ϕ— that simplifies the calculations without changing the values of E and B. In other words, the electric and magnetic fields remain invariant under “gauge transformations”—analogous to the way a rotating ball looks the same no matter which way you turn it. Gauges are now used to simplify calculations not just in electromagnetism, but in quantum theory and relativity, too.

向量分析:海维赛德和吉布斯的数学遗产

VECTOR ANALYSIS: THE MATHEMATICAL LEGACY OF HEAVISIDE AND GIBBS

我花了很多时间来理清关于麦克斯韦方程组作者的神话和现实,因为除了说明人们很难理解麦克斯韦的四元全矢量方程之外,它还表明了人们对科学发展方式经常存在的误解。再举个例子,牛顿定律从来没有以他在《数学原理》中写的形式写出,因为新的数学技术、符号和理解使其他人能够以现代矢量微积分形式写出它们。但内容仍然是牛顿的。麦克斯韦方程组也是如此——而且他比牛顿更接近现代形式。所以,海维赛德最重要的数学创新不是他对麦克斯韦方程本身的重新表述,而是他对现代矢量分析的发展。

I’ve spent time untangling the reality from the myth about who wrote Maxwell’s equations, because aside from illustrating how difficult it was for people to appreciate Maxwell’s quaternionic whole-vector equations, it shows something that is still often misunderstood about how science evolves. Newton’s laws, to take another example, are never written in the form he wrote them in Principia, because new mathematical techniques, notations, and understandings enabled others to write them in their modern vector calculus form. But the content is still Newton’s. And so it is with Maxwell’s equations—and he came much closer to the modern form than Newton did. So, Heaviside’s most important mathematical innovation is not so much his reformulation of Maxwell’s equations per se, as his development of modern vector analysis.

在他非凡著作《电磁理论》的第一卷中,海维赛德详细阐述了矢量分析的所有规则——将矢量从其作为四元数虚部的原始角色中完全解放出来。毕竟,他说,“我们生活在一个矢量的世界里”——正如我在第 3 章中指出的,牛顿是第一个阐明这一点的人。因此,亥维赛继续说道:“矢量代数或矢量语言是绝对必要的。” 16并且,在他关于电报和相关实用电磁理论的多篇论文中,他继续展示了这种语言的强大力量。当然,今天,你打开物理论文或教科书时一定会看到矢量。然而,在 19 世纪 90 年代初期,矢量分析要么是令人尊敬的分量法的后裔,要么是四元数的骄傲自大的弟弟 — — 这取决于你与谁交谈。汤姆森属于第一阵营,泰特属于第二阵营。

In the first volume of his extraordinary book Electromagnetic Theory, Heaviside set out in detail all the rules of vector analysis—breaking vectors away completely from their original role as the imaginary part of a quaternion. After all, he said, “we live in a world of vectors”—something Newton was the first to make clear, as I showed in chapter 3. So, Heaviside continued, “an algebra or language of vectors is a positive necessity.”16 And he went on to show, in his many papers on telegraphy and related practical electromagnetic theory, just how powerful this language could be. Today, of course, you can’t open a physics paper or textbook without seeing vectors. In the early 1890s, however, vector analysis was still either the cheeky offspring of the venerable component method or the cocky younger sibling of the quaternion—depending on whom you spoke to. Thomson was in the first camp, Tait in the second.

麦克斯韦和克利福德是最早从四元数转向向量分析的人。然而,尽管他们已不在人世,赫维赛德并不是唯一一个为向量而战的人。在大西洋彼岸,约西亚·威拉德·吉布斯也受到了麦克斯韦《数学分析》的启发。和赫维赛德一样,吉布斯对麦克斯韦的向量应用很感兴趣——与泰特不同,吉布斯对向量的兴趣并不在于它们的数学之美。事实上,他的研究生涯是从非常实用的方式开始的,他研究火车车厢的齿轮和刹车,以及蒸汽机的调速器——当时正值 19 世纪 60 年代,美国的铁路建设如火如荼。第一条横贯大陆的铁路线是一个庞大的基础设施项目,于 1869 年完工——但那时吉布斯已经将注意力从工程学转向物理学。事实上,他刚从欧洲回来,在那里待了三年,旁听了热力学这门新兴科学专家的讲座,在接下来的几年里,他为这门新兴科学做出了重大贡献。但当他在 1873 年读到麦克斯韦的《电磁论》时,他越来越被电磁物理学所吸引——因此,他对矢量语言也产生了浓厚的兴趣。

Maxwell and Clifford were the earliest to head away from quaternions toward vector analysis. Yet although they were now gone, Heaviside was not alone in his battle for vectors. Across the Atlantic, Josiah Willard Gibbs had also been inspired by Maxwell’s Treatise. And like Heaviside, Gibbs was interested in Maxwell’s applications of vectors—unlike Tait, Gibbs wasn’t interested in vectors for their mathematical beauty. Indeed, he had begun his research career in a very practical way, working on gears and brakes for railway carriages, and governors for steam engines—this was in the 1860s, when the building of railway lines in America was gathering pace. The first transcontinental line, a massive infrastructure project, was completed in 1869—but by then Gibbs had already turned his attention from engineering to physics. In fact, he’d just returned from three years in Europe, auditing lectures from experts in the new science of thermodynamics, and over the next few years he made major contributions to this new science. But when he read Maxwell’s Treatise in 1873, he became increasingly drawn to the physics of electromagnetism—and, therefore, to the language of vectors.

和海维赛德一样,吉布斯注意到麦克斯韦使用了标量和矢量的乘积,而不是完整的四元数乘积。这就是为什么在 19 世纪 80 年代,吉布斯和海维赛德都开始探索非四元数的矢量——当时他们还不知道对方的工作。

Like Heaviside, Gibbs noticed that Maxwell had made use of scalar and vector products, rather than the full quaternion product. Which is why, in the 1880s, Gibbs and Heaviside both branched out on their own exploration of vectors sans quaternions—without yet knowing of each other’s work.

虽然海维赛德是学术界的局外人,但吉布斯是一位年轻的教授,身处科学活动发生的世界另一端——他于 1871 年被任命为耶鲁大学数学物理学教授,但他与科学主流的地理距离也使他在某种程度上成为局外人。事实上,麦克斯韦几乎是当时,只有他一人认识到吉布斯早期热力学工作的重要性。但当海维赛德痛斥学术势利时,吉布斯却默默地继续自己的工作——1881 年和 1884 年,他自费出版了他的小书《矢量分析要素》第 1 部分和第 2 部分。

While Heaviside was an academic outsider, Gibbs was a young professor on the other side of the world from where the scientific action was taking place—he’d been appointed professor of mathematical physics at Yale in 1871, but his geographical distance from the scientific mainstream made him something of an outsider, too. In fact, Maxwell had been just about the only one, at the time, to recognise the importance of Gibbs’s early work on thermodynamics. But while Heaviside railed against academic snobbery, Gibbs quietly got on with his own work—and in 1881 and 1884, he self-published parts 1 and 2 of his little book Elements of Vector Analysis.

他并不是第一个研究“多重代数”的美国人。“多重代数”是指用符号表示多个数的代数,因此“双重代数”是复数,向量是“三重代数”(在三维空间中),四元数是“四重代数”,因为它们包含一个标量和三个空间向量分量,等等。受汉密尔顿发现一种没有交换律的新代数的启发,哈佛大学数学教授本杰明·皮尔斯长期致力于创建几十个其他新代数。后来,他的儿子查尔斯证明,唯一允许除法或逆代数(从而允许解方程和反转旋转的方式)的代数是实数(即传统的学校代数)、复数和四元数。本杰明和他的另一个儿子詹姆斯后来在哈佛大学讲授四元数——皮尔斯家族无疑具有数学天赋。

He wasn’t the first American to work on what he called “multiple algebras,” algebras whose symbols represent more than one number, so that “double algebras” were complex numbers, vectors were “triple algebras” (in 3-D space), quaternions were “quadruple algebras” because they included a scalar plus the three spatial vector components—and so on. Inspired by Hamilton’s discovery of a new algebra without the commutative law, the Harvard mathematics professor Benjamin Peirce had long been working on creating dozens of other new algebras. Later, his son Charles proved that the only algebras allowing division or the existence of inverses—and hence ways of solving equations and of reversing rotations—are those of real numbers (that is, traditional school algebra), complex numbers, and quaternions. Benjamin, and later another of his sons, James, gave lectures on quaternions at Harvard—the Peirces were certainly a mathematically talented family.

另一方面,吉布斯教授的是矢量分析课程,而不是四元数。正如我所提到的,他还给了我们标量和矢量积的点和叉符号——这让查尔斯·皮尔斯很不高兴。吉布斯回应了皮尔斯的批评,说他自己也不确定该使用哪种符号。但他跟随泰特和汉密尔顿使用希腊字母表示矢量;另一方面,海维赛德使用粗体表示矢量,但保留了汉密尔顿的SV表示标量和矢量积。所以,你可以看到现代符号——如上面四个麦克斯韦方程的现代形式——是海维赛德和吉布斯符号的综合。这是新概念的新符号需要一段时间才能找到最佳形式——或者至少是最受欢迎的形式——的另一个例子。

Gibbs, on the other hand, taught classes on vector analysis, not quaternions. He also gave us the dot and cross notation for scalar and vector products, as I’ve mentioned—much to Charles Peirce’s displeasure. Gibbs replied to Peirce’s criticism, saying he felt unsure himself about which notation to use. But he followed Tait and Hamilton in using Greek letters for vectors; Heaviside, on the other hand, used boldface type for vectors but kept Hamilton’s S and V for scalar and vector products. So, you can see how the modern notation—as in the modern form of the four Maxwell equations above—is a synthesis of both Heaviside’s and Gibbs’s symbols. It’s another example of the way new notations for new concepts take a while to find their best form—or, at least, their most popular one.

吉布斯从麦克斯韦的《数学论》开始研究向量,但几年后他发现了格拉斯曼的作品。和其他人一样,他似乎从未读过《数学论》,但他对格拉斯曼产品的通用性印象深刻——尤其是事实上,它们可以推广到三维以上。标量积也是可以推广的:如果两个向量ab各有n个分量,分别标记为a 1 , … , a nb 1 , … , b n,那么它们的标量积就是:

Gibbs got his start in vectors from Maxwell’s Treatise, but a few years later he discovered Grassmann’s work. Like everyone else, it seems, he never did manage to read all of Ausdehnungslehre, but he was mightily impressed with the generality of Grassmann’s products—especially the fact that they were generalisable to more than three dimensions. The scalar product is also generalisable: if two vectors a and b each have n components, labeled a1, … , an and b1, … , bn, then their scalar product is just:

a b = a 1 b 1 + a 2 b 2 + a 3 b 3 + …+ an b n

ab = a1b1 + a2b2 + a3b3 + … + anbn.

如今,这种一般形式通常被称为“内积”,这是格拉斯曼的观点。相比之下,我已经提到过,向量(或交叉)积仅在三维中定义,而格拉斯曼的类似“外积”是通用的,我们将在第 11 章中更详细地讨论这一点。

Today this general form is often called an “inner product,” following Grassmann. By contrast, I’ve already mentioned that the vector (or cross) product is only defined in three dimensions, whereas Grassmann’s analogous “outer product” was general, and we’ll see this in more detail in chapter 11.

吉布斯对格拉斯曼印象深刻,他写信给儿子小赫尔曼,敦促他出版父亲早期的作品——吉布斯似乎非常希望格拉斯曼应该优先于汉密尔顿。以至于在他的《矢量分析要素》的前言中,他承认了格拉斯曼和克利福德,但没有提到麦克斯韦,更不用说泰特和汉密尔顿了。我们知道他的主要贡献来自麦克斯韦的《矢量分析》 。17因此,吉布斯在否认应得的公共荣誉方面似乎相当吝啬。也许他想把矢量分析作为一门与四元数完全不同的新学科来呈现,因为我们很快就会看到他是如何坚定地捍卫这一议程的。尽管如此,格拉斯曼值得吉布斯的努力,这最终导致了他长期被忽视的文集的出版。

Gibbs was so impressed with Grassmann that he wrote to his son, Hermann Jr., urging him to publish his father’s earliest work—Gibbs seemed very keen that Grassmann should have priority over Hamilton. So much so that in the preface to his Elements of Vector Analysis he acknowledged Grassmann and Clifford, but made no mention of Maxwell, let alone Tait and Hamilton. We know that his primary debt was to Maxwell’s Treatise, though, because of a letter he wrote to a German colleague.17 So Gibbs seems rather ungenerous in denying public credit where it was due. Perhaps he wanted to present vector analysis as a new subject quite separate from quaternions, for we’ll see shortly how determinedly he defended this agenda. Still, Grassmann deserved Gibbs’s efforts, which ultimately led to the publication of his long-neglected collected works.

矢量“战争”

THE VECTOR “WARS”

1888 年,吉布斯将《向量分析》的副本寄给了他能想到的所有人,包括汤姆森、泰特和海维赛德。海维赛德很高兴地发现,他并不是唯一一个研究向量的人,但泰特却怒不可遏!到了 19 世纪 90 年代,《自然》杂志和其他领先的科学杂志上爆发了一场毫不文雅的争论。问题是:是否应该使用整个向量系统——汤姆森和亚瑟·凯莱等人认为不,分量就足够了;但如果使用它们,那么是使用汉密尔顿的四元数还是格拉斯曼的代数?还是应该使用吉布斯的以及海维赛德的功利主义矢量分析,泰特认为这只不过是抄袭汉密尔顿的吗?

In 1888, Gibbs sent a copy of his Vector Analysis to everyone he could think of—including Thomson, Tait, and Heaviside. Heaviside was pleased to find he was not the only one working on vectors—but Tait was infuriated! And by the 1890s, a none-too-genteel debate had erupted in the pages of Nature and other leading scientific magazines. The question was: Should wholevectorial systems be used at all—the likes of Thomson and Arthur Cayley said no, components were sufficient; but if they were to be used, then was it Hamilton’s quaternions or Grassmann’s algebra? Or should it be Gibbs’s and Heaviside’s utilitarian vector analysis, which Tait saw as little more than a rip-off of Hamilton?

吉布斯尤其激起了泰特的愤怒:“他称吉布斯的《矢量分析》为‘一种雌雄同体怪物’,由汉密尔顿和格拉斯曼的符号组合而成。”在 1891 年的《自然》杂志上,吉布斯非常有礼貌地反击了泰特的诽谤。(当时的《自然》杂志不像今天这样光鲜亮丽——当时的技术还不够——但到了 19 世纪 90 年代,它已经成为科学家们展示研究成果和讨论观点的热门场所。)后来,赫维赛德加入进来,暗示第一轮胜利属于吉布斯。后来,赫维赛德为了表明公平,表示争论的结局取决于你的观点。从四元数而非物理学的角度来看,他说:“泰特教授是对的,完全正确,四元数提供了一种独特、简单、自然的方法来处理四元数。”为了确保读者能理解这个笑话,他补充道:“注意强调。”但他确实继续谈到了四元数的数学之美,而不是他所看到的四元数在物理学中的“无用性” 。18

Gibbs, in particular, roused Tait’s ire: “a sort of hermaphrodite monster,” he called Gibbs’s Vector Analysis, “compounded of the notations of Hamilton and Grassmann.” In an 1891 Nature article, Gibbs fought back, very politely, against Tait’s slur. (Nature wasn’t the glossy magazine it is today—the technology wasn’t up to it—but by the 1890s it had become a very popular place for scientists to present their research and discuss their views.) Then Heaviside joined in, suggesting that round one had gone to Gibbs. Later, in a show of even-handedness, Heaviside said the resolution of the debate depended on your point of view. From the quaternionic rather than the physics viewpoint, he said, “Prof. Tait is right, thoroughly right, and Quaternions furnishes a uniquely simple and natural way of treating quaternions.” Just to make sure his readers got the joke he added, “Observe the emphasis.” But he did go on to speak of the mathematical beauty of quaternions, as opposed to what he saw as their “uselessness” in physics.18

泰特立即通过《自然》杂志回复了吉布斯。我已经提到过,既然叉积只在三维空间中定义,那么四元数积也是如此。因此,泰特决心不承认他钟爱的四元数有任何局限性,他对吉布斯强调将矢量扩展到三维以上的重要性提出了质疑,他反问道:“物理学家与三维以上的空间有什么关系?”这表明事后诸葛亮可以让历史变得更加有趣——虽然泰特在 1891 年是一位顶尖科学家,但仅仅 14 年后,年轻的爱因斯坦就证明了狭义相对论需要四维(t、x、y、z)——通常的三个空间维度加上一个时间坐标。然而,早在 1891 年,泰特就坚持三维——但四元数在紧凑性方面确实比矢量表达式更胜一筹,更不用说与分量方程相比了。19

Tait lost no time replying to Gibbs, through the pages of Nature. I’ve mentioned already that since the cross product is only defined in three dimensions, then so is the quaternion product. So, determined not to admit any limitations on his beloved quaternions, Tait challenged Gibbs’s emphasis on the importance of extending vectors to more than three dimensions, asking rhetorically, “What have students of physics, as such, to do with space of more than three dimensions?” Which goes to show how hindsight can make history even more interesting—for while Tait was a leading scientist in 1891, just fourteen years later young Einstein will show that the special theory of relativity needs four dimensions, (t, x, y, z)—the usual three of space plus a time-coordinate. Back in 1891, though, Tait was plumping for three dimensions—but he did score a point for quaternions in terms of their compactness compared even with vector expressions, let alone in comparison with component equations.19

正如我们在第 7 章中看到的那样,汤姆森在他们共同编写的前沿物理学教科书TT′中战胜了泰特,四元数就是从这本书中衍生出来的。放逐。后来汤姆森回忆起他们在这个话题上的争执,说:“我们为四元数斗争了 38 年。”他承认泰特“被汉密尔顿在这方面的天才的独创性和非凡之美所吸引”——然而,他说泰特永远无法举例说明如何通过使用四元数更容易地解决物理问题。当然,泰特和麦克斯韦一样,坚持认为如果你用四元数或全矢量形式写一个问题,会更容易看到物理现象——但正如汤姆森反复指出的那样,你仍然必须根据它们的分量来解方程。例如,你可能还记得在学校时试图找到一个斜扔到空中的球的轨迹。你必须使用牛顿第二定律F = m a,首先将矢量“分解”为其分量(如图 8.1所示),然后将牛顿定律应用于每个力分量——正如汤姆森所说的那样!20

As we saw in chapter 7, Thomson had won out over Tait in T and T′, their joint cutting-edge physics textbook from which quaternions were banished. Later Thomson recalled their fights on the topic, saying, “We have had a thirty-eight years war over quaternions.” He did acknowledge that Tait had been “captivated by the originality and extraordinary beauty of Hamilton’s genius in this respect”—and yet, he said, Tait could never give an example of how physical problems could actually be solved more easily with the use of quaternions. Of course, Tait, like Maxwell, maintained that if you wrote a problem in quaternion or whole-vector form it was easier to see the physics—but as Thomson repeatedly pointed out, you still have to solve the equations in terms of their components. For instance, you might remember from school trying to find the trajectory of a ball thrown obliquely into the air. You have to use Newton’s second law, F = ma, and first you “resolve” the vectors into their components (as in fig. 8.1), and then you apply Newton’s law to each force component—just as Thomson said!20

与此同时,吉布斯和泰特继续通过《自然》杂志进行辩论——吉布斯始终保持冷静——海维赛德在《电工》杂志上发表了他的文章。然后泰特的前学生亚历山大·麦克法兰也加入了进来——他现在住在德克萨斯州,在给美国科学促进会的一次演讲中,他提出,一种更完整的代数可能会发展出来,统一四元数和格拉斯曼代数。(他是对的,正如我们在克利福德的几何代数中看到的那样,以及我们将在张量分析中看到的那样。)他甚至尝试了他自己的矢量分析形式,使i、jk成为实数而不是虚数,但它不像吉布斯和海维赛德的那么简洁,而且从未流行起来。21

Meantime, Gibbs and Tait continued their argument via Nature—Gibbs never losing his cool—with Heaviside adding his piece in The Electrician. Then Tait’s former student Alexander Macfarlane joined in—he was now based in Texas, and in an address to the American Association for the Advancement of Science, he suggested that a more complete algebra will likely evolve, unifying quaternions and Grassmannian algebra. (He was right, as we saw with Clifford’s geometric algebra, and as we’ll see with tensor analysis.) He even attempted his own form of vector analysis, making i, j, and k real rather than imaginary, but it wasn’t as neat as Gibbs’s and Heaviside’s, and it never caught on.21

到 1892 年,争论已经蔓延到了澳大利亚,剑桥毕业生亚历山大·麦考利 (Alexander McAulay) 当时在墨尔本大学教授数学和物理。他在 1 月份的澳大利亚科学促进会会议上发表了一篇关于四元数的论文,该论文于 6 月份发表在《哲学杂志》上,他也在《自然》杂志的辩论中发表了自己的观点。他坚定地站在四元数论者的阵营,并将物理学家们对四元数分析能力缺乏认识归咎于麦克斯韦,因为麦克斯韦使用了矢量和标量积,而不是完整的四元数积。麦克法兰开枪了对此,麦克斯韦表示,问题不在于麦克斯韦,而在于四元数本身的局限性——尤其是标量积前面那个烦人的减号。就这样,四元数论者和矢量分析家们以不同程度的敏锐和克制力争论着各自的观点。

By 1892, the debate reached as far as Australia, where Cambridge graduate Alexander McAulay was teaching mathematics and physics at the University of Melbourne. He’d given a paper on quaternions at the January meeting of the Australian Association for the Advancement of Science, which was published in the Philosophical Magazine in June, and he also added his voice to the debates in Nature. He was firmly in the quaternionists’ camp, and he blamed Maxwell for the physicists’ lack of appreciation of the power of quaternion analysis, because Maxwell had used vector and scalar products rather than the full quaternion product. Macfarlane shot back at this, saying it wasn’t Maxwell who was to blame but the limitation of quaternions themselves—especially that annoying minus sign in front of the scalar product. And so it went, with quaternionists versus vector analysts arguing their cases with varying degrees of perspicacity and temperance.

图像

图 8.1。寻找抛射物的轨迹。球被抛出后,其初速度为v,作用于球的唯一力是向下的重力。因此,没有作用于水平方向的力分量。分别考虑这些分量,首先找到运动水平分量的值:取消牛顿第二定律中的质量项,并使用牛顿的点符号表示时间导数,我们得到:

FIGURE 8.1. Finding the trajectory of a projectile. Once the ball is thrown, with initial velocity v, the only force acting on it is the downward force of gravity. So, there’s no force component acting in the horizontal direction. Taking these components separately, first find the value of the horizontal component of motion: canceling the mass term in Newton’s second law, and using Newton’s dot notation for time derivatives, we have:

= 0

= C = v cos θ

x = vt cos θ。

= 0

= C = v cos θ

x = vt cos θ.

C是初速度水平分量的大小;从原点投影的第二个积分常数为零。)

(C is the magnitude of the horizontal component of the initial velocity; the second constant of integration is zero for projection from the origin.)

现在找到运动的垂直分量:

Now find the vertical component of motion:

ÿ = − g

= − gt + C 1 = − gt + v sin θ

  =22+  θ + 2

ÿ = −g

= −gt + C1 = −gt + v sin θ

 y =gt22+ vt sinθ + C2

(但对于从原点投影,C 2 = 0)。然后,您可以通过用x表示t并将其代入y方程来找到抛物线轨迹。

(but C2 = 0 for projection from the origin). You can then find the parabolic trajectory by expressing t in terms of x and substituting into the equation for y.

1893年,麦考利担任新成立的塔斯马尼亚大学的第一位数学和物理学讲师。同年,他出版了著作《四元数在物理学中的效用》,该书最初是 1887 年他为争取剑桥大学著名的史密斯奖而写的一篇雄心勃勃的论文。麦克斯韦曾在 1854 年分享过该奖项,但麦考利知道,他写一篇关于四元数的论文会毁掉自己的机会,因为剑桥大学很少有人能理解四元数。正如他在书的前言中所解释的那样,那些教授这门课程的剑桥教授将其“视为代数,但这不是汉密尔顿式的,汉密尔顿将四元数视为一种几何方法。”正如我在第 4 章中提到的,虽然德摩根和其他人对代数本身感兴趣,但汉密尔顿的目标是找到一种用于几何的代数语言——特别是用于描述三维空间中的旋转。这就是他一开始就想到他的新四元数代数的方式。

By 1893, McAulay had taken a position as the first lecturer in mathematics and physics at the newly established University of Tasmania. In the same year he published his book Utility of Quaternions in Physics, which he’d initially written in 1887 as an ambitious essay for Cambridge’s prestigious Smith’s Prize. Maxwell had shared the prize back in 1854, but McAulay had known he was blighting his chances with an essay on quaternions, which few at Cambridge understood. As he explained in the preface of his book, those Cambridge professors who did teach the subject treated it “as an algebra, but this [is] not Hamiltonian Hamilton looked upon Quaternions as a geometrical method.” As I mentioned in chapter 4, while De Morgan and others were interested in algebra for algebra’s sake, Hamilton’s goal had been to find an algebraic language for geometry—in particular, for describing rotations in three-dimensional space. That’s how he’d hit upon his new algebra of quaternions in the first place.

也许麦考利没有意识到,麦克斯韦在他的《数学论》中表达了同样的观点,即几何学的重要性——这与所讨论的物理意义密切相关。四元数分析,麦克斯韦称之为“矢量学说”,本质上是一种几何箭头语言——因此它关注的是空间中的实际点和线,而不是它们的坐标。这种几何方法允许独立于坐标选择来建模物理量——因此它揭示了它们的不变的物理性质。我们将在下一章中看到更多关于这一点的内容,但这也是麦克斯韦不遗余力地指出整个矢量而不是基于坐标的分量的重要性的原因。他对电磁场建模很感兴趣,使用磁力线的几何形状——就像铁屑的线——围绕磁铁、电荷和电流。同样,1915 年,爱因斯坦将著名地使用这种“矢量学说”的张量扩展来建模弯曲时空的几何形状。事后看来,我们很容易将数学和科学思想的发展视为理所当然,并认为从麦克斯韦到爱因斯坦、从矢量到张量的路径很简单。很难相信仅仅一个世纪前,数学竟然如此具有争议性。但在 1893 年,麦考利和他的同行们还远未完成。

Perhaps McAulay hadn’t realised that in his Treatise Maxwell had expressed this same view about the importance of geometry—which relates closely to the physical meaning in question. Quaternion analysis, which Maxwell called “the Doctrine of Vectors,” is, at base, a language of geometrical arrows—so it focuses on the actual points and lines in space, rather than on their coordinates. This geometrical approach allows physical quantities to be modeled independently of the choice of coordinates—so it brings out their invariant, physical properties. We’ll see more about this in the next chapter, but it’s why Maxwell took so many pains to point out the importance of whole vectors rather than coordinate-based components. He was interested in modeling the electromagnetic field, using the geometry of the lines of force—like the lines of iron filings—around magnets, electric charges, and electric currents. Similarly, in 1915 Einstein will famously use the tensor extension of this “doctrine of vectors” to model the geometry of curved space-time. In hindsight it’s easy to take for granted the development of mathematical and scientific ideas, and to assume an easy path from Maxwell to Einstein and from vectors to tensors. It can seem hard to believe that barely a century ago, mathematics, of all things, could have been so controversial. But in 1893 McAulay and his peers were far from done.

在序言中,麦考利甚至主张学生应该学习四元数而不是笛卡尔几何——他还补充说,鼓励学生学习这门学科的唯一方法是让其“物有所值”,正如他所说的那样,增加有关该学科的考试题目。情况并没有太大变化:大多数学生似乎都想学习如何通过考试,而不是如何思考。这就是我们学生面临的压力,也是我们教育和就业政策的局限性。22前辈麦克斯韦一样,麦考利清楚地明白,在教学生思考和训练他们通过考试和求职面试之间做出艰难的选择,他试图用他的书激励未来的学生,直接吸引他们,鼓励他们永远沉浸在四元数的“狂喜”中。“当你醒来时,”他继续说,“你会忘记[考试],随着时间的推移,你会陷入财务困境。”但麦考利的诱惑之歌继续说道,你得到的不是财富,而是“天赐的梦想记忆”,即“沉浸在”四元数中,这个记忆将使你“成为比百万富翁更快乐、更富有的人”。这是高尚的建议,尽管有些奇怪和不切实际!(像他所有的同事一样,麦考利在向学生讲话时也使用“男人”和“他”,因为在女性一般不允许获得学位的时代,这是常态。不过,麦考利确实有一位女权主义妻子——他于 1895 年与艾达·巴特勒结婚,她继续为女性接受高等教育和投票的权利以及性教育和计划生育而奋斗。)23

In his preface McAulay went so far as to advocate that students should study quaternions rather than Cartesian geometry—adding that the only way to encourage students to study the subject would be to make it “pay,” as he tellingly put it, by including more exam questions on the subject. Not much has changed: most students, it seems, want to learn how to pass exams rather than how to think. Such are the pressures on our students, and the limitations of our education and employment policies.22 Like Maxwell before him, McAulay clearly understood this problematic choice between teaching students to think and training them to pass exams and job interviews, and he sought to inspire future students with his book, appealing directly to them and encouraging them to immerse themselves timelessly in “the delirious pleasures” of quaternions. “When you wake,” he continued, “you will have forgotten [exams] and in the fullness of time will develop into a financial wreck.” But instead of wealth, McAulay’s siren song continued, you will possess “the memory of that heaven-sent dream” of being “steeped” in quaternions, a memory that will make you “a far happier and richer man than the millionest millionaire.” It’s noble if bizarre and impractical advice! (Like all his colleagues McAulay used “man” and “he” when addressing students, for it was the norm at a time when women generally weren’t allowed to take degrees. McAulay did have a feminist wife, though—he married Ida Butler in 1895, and she continued to fight for women’s rights to higher education and the vote, and for sex education and family planning.)23

赫维赛德私下告诉吉布斯,麦考利“似乎是一个非常聪明的家伙,他知道这一点,但有时表现得有点过头了。”至于泰特,他丝毫没有失去他尖刻的热情,他对麦考利支持四元数的书的评论被证明是《自然》杂志封面故事的完美点击诱饵——用一个尖锐的时代错误来形容。他写道:“在阅读了吉布斯和赫维赛德的作品后,再读麦考利的作品,真是令人振奋。”然后,泰特对自己的风格过度和人身攻击视而不见,但也许是有原因的,他对麦考利的修辞手法提出异议——他认为它们会让人们对四元数失去兴趣,而不是改变他们。尽管如此,他还是把麦考利看作一个一个拥有“真正的力量与独创性”的人,“他抓起汉密尔顿向所有人展示的神奇武器,立即冲进丛林去追捕大型猎物。” 24这样一个大男子主义的殖民主义隐喻在今天并不受欢迎!

Heaviside privately told Gibbs that McAulay “seems to be a very clever fellow, and he knows it and shows it a little too much sometimes.” As for Tait, he had lost none of his acerbic exuberance, and his review of McAulay’s pro-quaternion book proved perfect clickbait—to use a pointed anachronism—for a cover story in Nature. “It is positively exhilarating,” he wrote, to read McAulay after “toiling through the arid wastes” in the work of Gibbs and Heaviside. Then, with stunning blindness to his own stylistic excesses and personal attacks, but perhaps with some reason, Tait took issue with McAulay’s rhetorical flourishes—he thought they would put people off quaternions rather than convert them. Still, he saw in McAulay a man of “genuine power and originality,” who “snatches up the magnificent weapon which Hamilton tenders to all, and at once dashes off to the jungle on the quest of big game.”24 Such a macho, colonialist metaphor would not go down well today!

相比之下,吉布斯反驳了麦考利对四元数的辩护,他认为汉密尔顿本人掩盖了矢量方法的简单性和强大性——尤其是当他将标量和矢量积合并为一个四元数积时。与麦考利相反,吉布斯认为汉密尔顿实际上忽视了矢量的几何方面。谁会想到,今天使用矢量时,有这么多问题需要考虑——几何、经济性、符号、物理解释、计算的简易性——而且充满激情!

Gibbs, by contrast, countered McAulay’s defense of quaternions by suggesting that Hamilton himself had obscured the simplicity and power of the vectorial approach—especially when he’d joined scalar and vector products into a single quaternion product. And in contrast to McAulay, Gibbs felt that Hamilton had actually sidelined the geometrical aspect of vectors. Who would have thought, when using vectors today, that there’d been so many issues to think through—geometry, economy, notation, physical interpretation, ease of computation—and with so much passion!

争论仍在继续,新的参与者也加入进来——尤其是泰特以前的学生、未来的传记作者卡吉尔·诺特。随后,亥维赛又回到了争论中,在《自然》杂志上,他认为泰特和诺特对四元数的辩护毫无意义,并敦促麦考利放弃四元数。“四元数的平静与和平被打破了,”他得意洋洋地补充道。“四元数堡垒中一片混乱;警报响起,入侵者四处游走,向入侵者投掷石块、浇灌沸水。”他是对的,入侵的矢量分析学家最终获胜,任何本科数学或物理书籍都可以证明这一点。麦考利确实放弃了四元数:1905 年,他去领导塔斯马尼亚的第一个水力发电系统——世界上第一个水力发电系统之一。这是一个合适的结局,因为早些时候他已经将四元数应用于电磁学和流体动力学。25

The debate continued, with new players joining in—notably Tait’s former student and future biographer, Cargill Knott. Then Heaviside came back into the fray, deeming, in the pages of Nature, Tait’s and Knott’s defenses of quaternions irrelevant, and urging McAulay to give up quaternions. “The quaternionic calm and peace have been disturbed,” he added triumphantly. “There is confusion in the quaternionic citadel; alarms and excursions, and hurling of stones and pouring of boiling water upon the invading host.” He was right, and the invading vector analysts eventually won, as any undergrad maths or physics book testifies. And McAulay did give up quaternions: in 1905 he went off to spearhead Tasmania’s first hydroelectricity system—one of the first in the world. It was a fitting finale, for earlier he had applied quaternions to both electromagnetism and hydrodynamics.25

• • •

• • •

在所有这些愤怒中,吉布斯作为一个冷静而理智的人脱颖而出。他在《自然》杂志上发表的文章《四元数和向量代数》中,对这场争论的真正原因进行了精彩的总结,从自我和优先性的问题切入,直指数学物理学的古老目的——阐明物理量之间的“关系和运算”。尽管他坚定地站在向量分析一边,但他认为这没什么如此新颖——它只是表达人们几个世纪以来一直在发展的数学运算的最简单、最有用的方式。他说,我们应该向历史和我们的学生致敬,承认这一长串的贡献者——在这里,他确实包括了泰特,他称赞泰特发展了汉密尔顿的工作,并将其应用到汉密尔顿之后,无论他如何热情和忠诚地宣称不是这样。毕竟,吉布斯说,“现代思想的潮流太广阔了,即使是汉密尔顿的方法也无法限制它。”这是一种值得记住的历史观——新思想的杰出发起者很少能把这些思想保持完美的形式,许多其他才华横溢的人也开始着手澄清、应用和扩展这些思想。26

Gibbs stands out in all this fury as a man of calm and reason. In his Nature article “Quaternions and the Algebra of Vectors,” he gave a brilliant summary of what the fuss was really about, cutting through questions of ego and priority to the age-old purpose of mathematical physics—that of clarifying the “relations and operations” between physical quantities. And although he was firmly on the side of vector analysis, he saw it as nothing so terribly new—it was just the simplest and most useful way of expressing the kinds of mathematical operations people had been developing for centuries. We owe it to history, he said, and to our students, to acknowledge this long line of contributors—and here he did include Tait, whom he praised for developing and applying Hamilton’s work beyond Hamilton, no matter how zealously and loyally he proclaimed otherwise. After all, said Gibbs, “the current of modern thought is too broad to be confined by the [methods] even of a Hamilton.” It’s a view of history that is worth remembering—that the dazzling initiators of new ideas rarely leave those ideas in perfect form, and that many other brilliant minds set to work clarifying, applying, and extending them.26

回到起点……

BACK TO THE BEGINNING …

事实上,“矢量战争”的历史可以追溯到有记载的数学之初。因为从本质上讲,这场世纪末的激烈争论是关于信息的最佳表示方式——以及用信息进行计算的最佳方式。美索不达米亚人和埃及人一直在研究如何用表格和数组记录地块大小、为开挖运河所要挖的土方量以及挖掘所需的金钱和劳动力成本等信息。他们通过查阅乘法表、算法摘要和毕达哥拉斯三元组列表(如序言中图 0.1所示的美索不达米亚泥板 Plimpton 322 中的列表)来计算这些信息。

In fact, the history relevant to the “vector wars” goes right back to the beginning of recorded mathematics. For at its core, this fiery fin-de-siècle debate was about the best way to represent information—and the best way to calculate with it. Representation is what the Mesopotamians and Egyptians had been working on with their tables and arrays for recording information about the size of land parcels, or the volume of earth to be dug for a canal and the cost in money and labour of the digging. And they calculated this information by consulting multiplication tables, summaries of algorithms, and lists of Pythagorean triples such as that in Plimpton 322, the Mesopotamian tablet shown in figure 0.1 in the prologue.

一千年后,古希腊人尝试过不仅表示商业和社会数据,还表示空间位置信息的想法——正如我们所见,他们的坐标概念发展成为我们熟悉的数学笛卡尔和极坐标网格。快进到 19 世纪 90 年代,你可以看到矢量/四元数/分量的争论是关于表示利用空间物理量信息的最佳方式。

A thousand years later, the ancient Greeks had experimented with the idea of representing not only commercial and sociological data but information about locations in space—and as we’ve seen, their idea of coordinates developed into our familiar mathematical Cartesian and polar grids. Fast-forward to the 1890s, and you can see that the vector/quaternion/ components debate was about the best way to represent and utilise information about physical quantities located in space.

正如 Tait 和其他四元数论者在 19 世纪 90 年代所论证的那样,四元数在信息紧凑表示方面优于向量,而四元数和整个向量都优于分量列表。但正如 Thomson一直坚持到最后,既然你无论如何都要用组件来计算,那为什么要费心为整个实体添加特殊符号呢?他并不是唯一一个持这种观点的人——凯莱就是其中之一。但如果他们生活在数字时代,他们肯定会改变想法,因为在数字时代,表示经济在成本、时间和能源方面都很重要。

As Tait and the other quaternionists argued in the 1890s—quaternions beat vectors for compact representation of information, while both quaternions and whole vectors beat lists of components. But as Thomson maintained till the very end, since you have to calculate with components anyway, why bother with special notation for the entities as a whole? He was not alone—Cayley, for one, had the same view. But they would surely have changed their minds had they lived in the digital age, where representational economy matters in terms of cost, time, and energy.

当然,泰特和麦克斯韦长期以来一直在从物理角度为四元数和整个向量辩护,但新的向量分析方法(其实数基向量i、j、k而不是四元数虚数i、j、k)对物理学家来说更为自然。因此,到 20 世纪初十年,很明显向量分析确实赢了——至少暂时如此。四元数必须等到计算机时代才能卷土重来——如今,新的研究论文定期出现,将四元数扩展并应用于航空航天导航、生物识别、机器人技术、分子动力学、控制系统、彩色图像处理等。

Of course, Tait and Maxwell had long been arguing for quaternions and whole vectors on physical grounds, but the new vector analysis approach— with its real basis vectors i, j, k rather than the quaternionic imaginary numbers i, j, k—proved more natural to physicists. And so, by the opening decade of the twentieth century, it was clear the vector analysts had, indeed, won—at least for the time being. Quaternions would have to wait for the computer age before they could make a comeback—and today new research papers appear regularly, extending and applying quaternions to aerospace navigation, biometric recognition, robotics, molecular dynamics, control systems, colour image processing, and more.

超越载体

BEYOND VECTORS

正是物理学,尤其是麦克斯韦理论,将向量方法带入主流——而物理学家吉布斯和电工物理学家海维赛德在 19 世纪 90 年代引领了向量取代四元数的潮流。但正如我们将在下一章中看到的那样,正是数学家发现了四元数和整个向量的关键特性,这首先导致了四维向量分析,然后是n维向量,最终是张量分析。不过,首先,我们将了解一下凯莱和泰特以及他们的数学家对向量之战的看法。

It was physics, especially Maxwell’s theory, that brought vector methods into the mainstream—and it was the physicist Gibbs and the electricianphysicist Heaviside who, in the 1890s, had led the charge for vectors over quaternions. But as we’ll see in the next chapter, it was mathematicians who identified the key qualities—in both quaternions and whole vectors— that led first to four-dimensional vector analysis, then to n-dimensional vectors, and ultimately to tensor analysis. First, though, we’ll check in with Cayley and Tait and their mathematicians’ view of the vector wars.

(9)从空间到时空

(9) FROM SPACE TO SPACE-TIME

矢量的新变化

A New Twist for Vectors

自 1863 年起,亚瑟·凯莱就担任剑桥大学萨德莱数学教授,事实上,他是第一个担任这个享有盛誉的教席的人,这个教席是由玛丽·萨德莱夫人捐赠的。人们对她知之甚少,只知道她热衷于慈善事业,剑桥的数学教席远非她唯一的慈善事业。有这样一位慈善家,凯莱很自然地成为女子高等教育的早期支持者。尽管她们不能在剑桥获得正式学位,但女性可以学习与男性类似的课程,1869 年和 1871 年,第一所女子住宿学院——格顿学院和纽纳姆学院成立。格顿学院的其中一位年轻女性是格蕾丝·奇泽姆,她觉得凯莱非常热情,但他对数学的态度却非常古板。她回忆起他“背对着听众站着,一边在黑板上写字,一边说话”时,学士服的“袖子”在飘扬。1但这只是一次性的评论:她需要得到格顿女校长和凯利的特别许可,才能去听他的课;除了少数例外,女生都在自己的学院里跟随导师学习,很少和男生一起听课。

Since 1863 Arthur Cayley had held the Sadleirian professorship of mathematics at Cambridge—in fact, he was the first to hold this prestigious chair, which had been endowed by Lady Mary Sadleir. Not much is known about her, except that she was devoted to good causes—the maths chair at Cambridge was far from her only beneficence. It is fitting, given such a benefactor, that Cayley was an early supporter of women’s higher education. Although they couldn’t take formal degrees at Cambridge, women were allowed to study lectures similar to those offered the men, and in 1869 and 1871 the first women’s residential colleges were established—Girton and Newnham. One of these young Girton women was Grace Chisholm, who found Cayley very welcoming, yet stiflingly old-fashioned in his approach to mathematics. She recalled the “flapping sleeves” of his academic gown “as he stood with his back to the listeners chalking and talking at the same time at the blackboard.”1 But this was a one-off comment: she’d needed special permission from the headmistress of Girton, as well as from Cayley, even to attend his class; with few exceptions, women studied with tutors in their own colleges, rarely attending lectures with the men.

1893 年,奇泽姆获得了相当于一等学位的学位。在接下来的几十年里,一项正式授予女性学位的动议被多次否决——包括 1897 年,当时一些男本科生为保留自己的特权而得意忘形,他们在镇上疯狂庆祝,造成了相当于十多万美元的损失。凯利一定很反感,因为他在 19 世纪 80 年代一直担任纽纳姆学院的理事会主席,也曾在格顿学院任教。弗吉尼亚·伍尔夫于 1928 年在这些学院发表了她著名的“自己的房间”讲座。2

Chisholm qualified for the equivalent of a first-class degree in 1893. A motion to formalise degrees for women was defeated, several times, over the next few decades—including in 1897, when some male undergrads were so chuffed at retaining their privilege they went on a wild celebratory rampage through town, causing the equivalent of more than a hundred thousand dollars’ worth of damage. Cayley must have been disgusted, for he’d been chairman of Newnham’s council through the 1880s, and he had also taught at Girton. Virginia Woolf would give her famous “Room of One’s Own” lectures at these colleges in 1928.2

在学生疯狂争论之前的几年里,矢量之争一直在上演,正如我们所看到的,物理学家们对现代矢量分析的支持正在取得进展。然而,在幕后,凯莱和彼得·泰特一直在从数学的角度悄悄地争论这个问题。早在 1888 年,凯莱就对泰特大声说:“我们是不可调和的,而且会一直如此。”但在 1894 年夏天——当时凯莱 73 岁,泰特比他年轻 10 岁——这两位英国数学界的元老公开发表了他们数学家对矢量争论的看法。在泰特的建议下,他们一起做了这件事,每个人都在爱丁堡皇家学会面前宣读了一篇论文。3

In the years just before that wild student rampage, the vector wars had been playing out, and as we saw, the physicists’ argument in favour of modern vector analysis was gaining ground. In the background, though, Cayley and Peter Tait had been quietly debating the issue from a mathematical point of view. Back in 1888 Cayley had exclaimed to Tait, “we are irreconcilable and shall remain so,” but in the summer of 1894—when Cayley was seventy-three and Tait ten years younger—these two elder statesmen of British mathematics went public with their mathematicians’ view of the vector debate. And at Tait’s suggestion, they did it together, each reading a paper before the Royal Society of Edinburgh.3

不变性的美丽概念

THE BEAUTIFUL CONCEPT OF INVARIANCE

凯莱以圆滑的语气开始了他的演讲,引用了泰特对四元数优势的看法:“它们给出了它们所应用问题的最一般陈述的解决方案,完全不受选择特定坐标轴的限制。”泰特的意思是,如果你改变参考系(比如旋转坐标轴),空间中的点将具有不同的坐标,向量(和四元数)将具有不同的分量。但正如你在图 9.1中看到的,当参考系旋转 θ 角时,向量的长度不会改变——它只是同一个向量——因此任何两个向量的标量和向量积也将保持不变。

Cayley began his presentation diplomatically, quoting Tait’s view of the advantage of quaternions: “They give the solution of the most general statement of the problem they are applied to, quite independent of any limitations as to the choice of particular coordinate axes.” What Tait meant was that if you change your frame of reference—by rotating the axes, say— your points in space will have different coordinates, and your vectors (and quaternions) will have different components. But as you can see in figure 9.1, the length of a vector doesn’t change when the frame is rotated through an angle of θ—it’s just the same vector—and so the scalar and vector products of any two vectors will remain the same, too.

这种非凡的性质被称为“不变性”,即即使从不同的轴测量,某些事物的成分也保持不变。两组坐标之间的关系称为一种“坐标变换”。坐标变换有两种基本类型:一种是简单的“变量变换”,例如从笛卡尔坐标系变换到极坐标系,如图 2.3所示,轴保持不变;另一种是框架之间的变换,即轴本身发生变化。(更专业地说,框架是一把尺子、一个时钟、和坐标系,使观察者能够协调时间和空间中的事件。)第二种坐标变换在许多实际问题中非常重要。

This remarkable property, where certain things stay the same even when the components are measured from different axes, is called “invariance.” The relationship between the two sets of coordinates is called a “coordinate transformation.” There are two basic types of coordinate transformation: one is simply a “change of variables,” such as transforming from Cartesian to polar coordinates as in figure 2.3, where your axes stay the same; the other is a transformation between frames—that is, where the axes themselves are changed. (More technically, a frame is a ruler, a clock, and a coordinate system, which allow the observer to coordinatize events in time and space.) It’s this second type of coordinate transformation that is important in many practical problems.

图像

图 9.1。向量a在通常的x - y坐标系中有分量 ( a 1 , a 2 ) ,在旋转的x′ - y′坐标系中有分量 ( a 1′ , a 2′ )。这是同一个向量,但有两个不同的坐标表示。从图中的几何形状可以看出,向量的长度在两个坐标系中都相同。从数学上讲,它在旋转下是不变的。对于第二个向量b来说也是如此。

FIGURE 9.1. The vector a has components (a1, a2) in the usual x-y frame, and (a1′, a2′) in the rotated x′-y′ frame. It is the same vector, with two different coordinate representations. The length of the vector is the same in both frames, as you can see from the geometry of the figure. Mathematically speaking, it is invariant under rotations. The same will be true for a second vector b.

因此,标量和矢量积在旋转下也是不变的,正如你使用这些积的几何定义所看到的那样:ab = ab cos Φ,其中ab是两个矢量的长度,Φ 是它们之间的角度;因为矢量不会改变,所以它们之间的角度不受坐标变化的影响。同样,a × b的大小(长度)为ab sin Φ,方向垂直于两个矢量的平面,当你以这种方式旋转轴时,这个平面(此处图表中的页面平面)不会改变。当轴像这样改变时,“坐标变换方程”为:

So the scalar and vector products are invariant under rotations, too, as you can see by using the geometric definitions of these products: ab = ab cos Φ, where a and b are the lengths of the two vectors, and Φ is the angle between them; because the vectors don’t change, the angle between them isn’t affected by the coordinate change. Similarly, a × b has a magnitude (length) of ab sin Φ, and a direction perpendicular to the plane of the two vectors, and this plane—the plane of the page in the diagram here—doesn’t change when you rotate the axes in this way. When the axes are changed like this, the “coordinate transformation equations” are:

x′ = x cos θ + y sin θ,y′ = − x sin θ + y cos θ。

x′ = x cos θ + y sin θ, y′ = −x sin θ + y cos θ.

我们在图 4.2中看到了一个机械臂的例子(我们旋转的是手臂而不是轴)。但关键点不是细节,而是当你改变坐标系时,两组坐标通过特定的方程相互关联。这是张量概念的关键。

We saw an example of this for the robot arm in figure 4.2 (where we rotated the arm rather than the axes). But the key point is not the details but the fact that when you change your coordinate frame, the two sets of coordinates are related by specific equations. This is key to the idea of tensors.

由于标量积等整体向量表达式在某些坐标变换下是不变的,因此在表示向量时,“无坐标”或“坐标独立”是“整体”的更数学的说法。1905 年的狭义相对论将充分说明不变性在物理学中的重要性(以及如何),但在 1894 年,当 Tait 和 Cayley 争论这个问题时,他们关注的是数学而不是物理学。

Because whole-vector expressions such as scalar products are invariant under certain coordinate transformations, “coordinate-free”—or “coordinate-independent”—is a more mathematical way of saying “whole” when speaking of representing vectors. The 1905 special theory of relativity will show in spades just why (and how) invariance matters in physics, but in 1894, when Tait and Cayley were debating the issue, they were focusing on maths rather than physics.

事实上,不变性的概念不仅在数学上令人着迷,而且其应用范围比在物理学中更为广泛。首先来看一个现代数字应用,神经网络通过将信息从一个节点或“神经元”传递到另一个节点或“神经元”来处理复杂数据,就像我们大脑中的神经元一样。在每个节点,模型(类似于线性回归)为输入数据分配权重,根据每段数据在所需输出中的重要性对其进行加权。我们在第 4 章的搜索引擎排名算法中看到了类似的情况。然而,在神经网络中,每一层节点都会增加模型的复杂性,因此在计算将一层神经元映射到下一层的数学运算时,程序员必须确保信息的关键特征在这些“映射”或坐标变换下是不变的。

In fact, the idea of invariance is both mathematically fascinating and more broadly applicable than in physics alone. To take a modern digital application first, neural networks handle complex data by passing information from one node or “neuron” to another, just as the neurons in our brains do. At each node, a model (analogous to linear regression) assigns weights to the input data, weighting each piece of data according to its importance in the desired output. We saw something similar in chapter 4, with search engine ranking algorithms. In neural networks, though, each layer of nodes adds more complexity to the model, so when working out the maths for mapping one layer of neurons onto the next, programmers must make sure that key features of the information are invariant under these “maps” or coordinate transformations.

例如,2022 年,DeepMind AI 团队的“AlphaFold”神经网络成功预测了几乎所有已知蛋白质的结构——其中有两亿种蛋白质是多年来通过对各种物种的基因测序而确定的。蛋白质是折叠成三维形状的氨基酸链——形状决定了它们的功能。因此,科学家希望了解这种形状,以便创造新型药物、用于农业或污染控制的新酶,或通过其相关蛋白质检测 SARS-CoV-2 病毒中值得关注的新变体,等等——只是这些氨基酸链的折叠方式太多了,长期以来一直无法弄清楚实际结构。AlphaFold该算法使用每种蛋白质氨基酸的线性(1-D)序列以及有关相关蛋白质已知结构的训练数据来预测折叠蛋白质中所有关键原子的 3-D 坐标。4它从一层节点进展到下一层节点时,这种算法会从这些输入数据中“学习”更多关于蛋白质可能形状的知识,因此程序员必须确保关键属性(例如原子之间的距离)在信息传输到下一个节点的坐标系时保持不变。

For instance, in 2022 the DeepMind AI group’s “AlphaFold” neural network succeeded in predicting the structures of virtually all the known proteins—two hundred million of them, which had been identified over the years through genetic sequencing of various species. Proteins are chains of amino acids that fold into three-dimensional shapes—and it’s the shape that governs their function. So, it’s the shape that scientists want to understand in order to create new types of drugs, or new enzymes for agriculture or pollution control, or to detect new variants of concern in the SARS-CoV-2 virus via its associated proteins, and so on—except that there are so many possible ways these chains of amino acids can fold that it had long been impossible to figure out the actual structures. The AlphaFold algorithm used the linear (1-D) sequence of each protein’s amino acids, along with training data about known structures of related proteins, to predict the 3-D coordinates of all the key atoms in the folded protein.4 As it progresses from one layer of nodes to the next, such an algorithm “learns” more about the protein’s possible shape from this input data, so programmers have to make sure that key attributes, such as the distance between the atoms, stay the same when the information is transferred to the coordinate frame of the next node.

与其他分子建模以及计算机视觉应用一样,蛋白质算法也必须学会识别正确的形状,即使形状被旋转了——这意味着 3D 形状的数学表示在旋转时必须是不变的。不变性的概念在许多神经网络和其他技术应用中都很重要,从卫星图像和生物医学显微镜图像到保持詹姆斯·韦伯太空望远镜的位置。

As with other molecular modeling, and also with computer vision applications, the protein algorithm also had to learn to recognise the correct shape even if it is rotated—which meant that the mathematical representation of the 3-D shape had to be invariant under rotations. The concept of invariance is important in many neural network and other technological applications, from satellite imagery and biomedical microscopy imagery to keeping the James Webb Space Telescope in place.

凯莱和其他不变性数学的先驱们会对这些复杂的现代应用感到震惊。另一方面,他们知道各种各样的东西都可以是不变的。例如,在第 4 章中我们看到,如果将一本书水平旋转 90°,然后垂直翻转 180°,当你反向执行这些操作时,它的方向会有所不同——而如果你以这种方式旋转一个毫无特征的盒子或球,它每次看起来都会一样。这是因为盒子和球是对称的。同样,雪花通常有六个点或角,而且它非常对称——所以当你将它旋转 60° 的倍数时,它看起来是一样的。 (说得更严谨些,自然界中并非所有的雪花都是完全对称的 - 但你可以看出这一点。)换句话说,它在这些旋转下是不变的 - 而且在绕其对称轴 180° 旋转或反射下也是不变的。

Cayley, and the other pioneers of the maths of invariance, would be stunned at these sophisticated modern applications. On the other hand, they knew that all sorts of things can be invariant. For example, in chapter 4 we saw that if you rotated a book horizontally through 90° and then flipped it over vertically through 180°, it would have a different orientation when you performed these operations in reverse—whereas if you rotated a featureless box or ball in this way, it would look the same each time. That’s because the box and ball are symmetrical. Similarly, a snowflake generally has six points or corners, and it is beautifully symmetrical—so it looks just the same when you rotate it through multiples of 60°. (To be pedantic, in nature not all snowflakes are perfectly symmetrical— but you can see the point.) In other words, it is invariant under these rotations—and also under 180° rotations, or reflections, about its axes of symmetry.

因此,不变性与对称性相关——在数学中这两个词经常互换使用。

So, invariance is related to symmetry—and in maths these two words are often used interchangeably.

至于 Tait,他对矢量和四元量的不变性很感兴趣,并给出了ab = 0 的例子,这个方程保持即使你在不同的坐标系中测量向量的分量,如图9.1所示,情况也是如此。这是因为标量积本身在这些坐标变换下是不变的。你可能还记得在学校里,等式ab = 0 可以解释为向量ab彼此垂直。垂直是几何性质,向量是四元数(的一部分)——因此 Tait 认为整个向量和四元数可以给出清晰、直接的几何解释。

As for Tait, he was interested in the invariance of vector and quaternion quantities, and he gave the example of ab = 0, an equation that stays true even if you measure the vectors’ components in different coordinate frames, such as those shown in figure 9.1. That’s because the scalar product itself is invariant under these coordinate transformations. You may remember from school that the equation ab = 0 can be interpreted as saying the vectors a and b are perpendicular to each other. Being perpendicular is a geometric property, and a vector is (part of ) a quaternion—so Tait argued that whole vectors and quaternions give clear and immediate geometrical interpretations.

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图 9.2。雪花的对称性。图 18 摘自 Wilson A. Bentley 的《1901-2 年冬季雪晶研究,以及前几个冬季收集的附加数据》,《月度天气评论》第 30 卷,第 13 期(1903 年):第 607-16 页,https://doi.org/10.1175/1520-0493-30.13.607。

FIGURE 9.2. The symmetry of snowflakes. Plate 18 from Wilson A. Bentley, “Studies among the Snow Crystals during the Winter of 1901–2, with Additional Data Collected during Previous Winters,” Monthly Weather Review 30, no. 13 (1903): 607–16, https://doi.org/10.1175/1520-0493-30.13.607.

而凯莱则对使用不变性来帮助纯数学中解方程很感兴趣。代数不变量的一个简单例子是判别式,在我们在学校学习的二次方程中,它是b 2 − 4 ac 。它是一般二次方程解公式ax 2 + bx + c = 0中平方根符号下的表达式,正如你在下一个尾注中看到的那样,如果你通过以某种方式改变坐标来改变方程,它仍然保持不变——例如,用x′ = x + h替换x。换句话说,判别式在坐标变换x′ = x + h下是不变的。(正如我之前指出的,坐标变换只是一组方程,显示如何将原始框架中的坐标与新框架中的坐标联系起来——在这种情况下,当轴水平平移距离h时。)5

Cayley, on the other hand, was interested in using invariance to help solve equations in pure maths. A simple example of an algebraic invariant is the discriminant, which, in the quadratic case we learn at school, is b2 − 4ac. It’s the expression under the square root sign in the formula for the solution of the general quadratic equation, ax2 + bx + c = 0, and as you can see in the next endnote, it stays the same if you change the equation by changing the coordinates in certain ways—for example, by replacing x with x′ = x + h. In other words, the discriminant is invariant under the coordinate transformation x′ = x + h. (As I indicated earlier, a coordinate transformation is just a set of equations showing how to relate the coordinates in the original frame with those in the new one—in this case, when the axes are horizontally translated by a distance h.)5

对于矢量之战的最后一轮来说,这一切意味着,尽管两人都在探索不变性的概念,但凯莱对坐标和坐标变换感兴趣,而泰特对整个四元数和矢量感兴趣。你可以看到为什么他们在表示矢量信息的最佳方式上仍然“无法调和”!

What this all means for the final round of the vector wars is that while both men were exploring the idea of invariance, Cayley was interested in coordinates and coordinate transformations, whereas Tait was interested in whole quaternions and vectors. You can see why they remained “irreconcilable” on the best way of representing vectorial information!

泰特不知道,他所倡导的无坐标、不变的方程式书写方式,如ab = 0,是向量分析和张量分析之间的关键联系。然而,凯莱的回应与威廉·汤姆森一样,他说计算仍然需要坐标。因此,他在 1894 年对爱丁堡皇家学会的演讲中宣称,就像满月比昏暗的月光更美丽一样,“所以我认为四元数的概念比它的任何应用都更美丽。”为了强调这一点,他补充说,四元数公式就像一张袖珍地图,一旦展开,就会非常有用——一旦你将公式转换成基于坐标的分量。几年前,当奇泽姆第一次见到凯莱时,她觉得“据说曾经在伟大数学家眼中闪耀的火焰已经消失了。”但凯利在矢量战争中的热情仍然足够高涨。6

Tait didn’t know it, but the coordinate-free, invariant way of writing equations that he championed, such as ab = 0, is the key link between vector analysis and tensor analysis. Cayley, however, responded just like William Thomson, saying you still need coordinates to do the calculations. So, he proclaimed—in his 1894 address to the Royal Society of Edinburgh—that just as the full moon is more beautiful than a dimmer moonlit view, “so I regard the notion of a quaternion as far more beautiful than any of its applications.” To emphasise the point, he added that a quaternion formula was like a pocket map, incredibly useful once you unfold it—once you translate the formula into its coordinate-based components. When Chisholm had first met Cayley just a couple of years earlier, she felt that “The fire was gone that they say had once gleamed from the eyes of the great mathematician.” But there was still fire enough in Cayley’s engagement with the vector wars.6

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彼得·格思里·泰特 (Peter Guthrie Tait) 在他的研究中展示了电的物理原理;经爱丁堡詹姆斯·克拉克·麦克斯韦基金会慷慨许可使用。

Peter Guthrie Tait in his study, demonstrating the physics of electricity; used with generous permission from the James Clerk Maxwell Foundation, Edinburgh.

泰特不愿被文学类比所打败,他回答说,与展开的袖珍地图不同,基于坐标的几何学就像一把蒸汽锤,需要专家操作才能发挥作用,而不是破坏性。另一方面,四元数是如此通用,以至于“就像大象的鼻子,随时准备应对任何事情”,无论大小——用他的话来说,捡起面包屑或勒死老虎。我们通常不会把夸张与数学家联系起来——但泰特笑得很开心,眼睛闪闪发光,有点像个恶作剧者。7

Not to be outdone via literary analogies, Tait responded that rather than an unfolded pocket map, coordinate-based geometry was like a steam hammer, requiring expert manipulation so it would be useful rather than destructive. Quaternions, on the other hand, were so general they were “like an elephant’s trunk, ready at any moment for anything,” large or small— picking up a breadcrumb or strangling a tiger, as he put it. We don’t usually associate hyperbole with mathematicians—but Tait, with his hearty laugh and twinkling eyes, was a bit of a prankster.7

• • •

• • •

关于无坐标矢量和四元数方程与基于坐标的分量方程的相对优势的争论持续了在接下来的十年里。这是一场英语辩论,部分原因是格拉斯曼的矢量系统当时还没有真正流行起来,而四元数在英国以外几乎不为人知——而且年轻的矢量分析异军突起的吉布斯和海维赛德也是英语国家。但历史学家迈克尔·克罗提出了这些关于矢量最佳符号长期争论的另一个原因。英国数学家意识到莱布尼茨的微积分符号比牛顿的更适合计算,到 19 世纪初,这种符号优势帮助欧洲大陆数学超越了英国。所以,他们不希望同样的事情发生在汉密尔顿的四元数和矢量上。8

Sparring over the relative advantages of coordinate-free vector and quaternion equations versus coordinate-based component ones continued for the next decade. It was an English-speaking debate, partly because Grassmann’s vectorial system hadn’t really taken off at that time, while quaternions were still virtually unknown outside Britain—and the young vector analysis mavericks Gibbs and Heaviside were English-speakers, too. But historian Michael Crowe suggests another reason for these long-running debates over the best notation to use for vectorial quantities. British mathematicians were aware that Leibniz’s symbolism for calculus was much better suited to calculations than was Newton’s, and that by the early nineteenth century this notational advantage had helped continental mathematics to leap ahead of that in Britain. So, they did not want the same thing to happen with Hamilton’s quaternions and vectors.8

不管出于什么原因,我之所以强调这个看似晦涩难懂的争论,是因为它还表明,即使是最优秀的数学家,也很难理解向量在整体、无坐标形式下的价值,而不是其分量形式。然而,数学家会把向量变成张量(或者更确切地说,会识别和概括向量背后的不变性和代数结构)。因此,尽管物理学家在创建向量分析方面处于领先地位,但数学家为爱因斯坦的物理学杰作——广义相对论(张量)理论铺平了道路。

Whatever the reason for it, I’m emphasising this seemingly arcane dispute because it also shows how difficult it was for even the best mathematicians to appreciate the value of vectors in their whole, coordinate-free form, as opposed to their component forms. Yet it is mathematicians who will turn vectors into tensors (or rather, who will identify and generalise the invariance and algebraic structure underlying vectors). So, although the physicists were ahead of the game in the creation of vector analysis, it is mathematicians who will pave the way for Einstein’s masterpiece of physics, the (tensor) theory of general relativity.

与此同时,向量故事的下一步开始于小爱因斯坦和他在苏黎世联邦理工学院的数学教授赫尔曼·闵可夫斯基。1900 年,爱因斯坦在“理工学院”(这是该学院的昵称)完成了学业,但他是那一小群毕业生中唯一一个没有在那里获得工作机会的人,就像麦克斯韦毕业时没有获得剑桥大学的奖学金一样。麦克斯韦被认为太粗心了,而年轻的爱因斯坦却过于自信,他的教授们不喜欢他。(他也是犹太人,所以反犹太主义可能是他失业的原因之一:爱因斯坦本人也这么认为。9 因此,在经济拮据、需要照顾怀孕的未婚妻的情况下,他在瑞士专利局接受了那份传奇的工作。正是在那几年里,他利用业余时间发展了自己的狭义相对论。

Meantime, the next step in the story of vectors begins with the younger Einstein and his maths professor Hermann Minkowski, at the Federal Polytechnic School in Zurich. In 1900 Einstein completed his degree at the “Poly,” as it was affectionately known—but he was the only one of his small class of graduates not to be offered a job there, just as Maxwell hadn’t been offered a fellowship at Cambridge when he graduated. Maxwell had been deemed too careless, while young Einstein was far too sure of himself for his professors’ liking. (He was also Jewish, so anti-Semitism may have played a role in his lack of employment: Einstein himself thought so.9) And so, in desperate financial straits and with a pregnant fiancée to care for, he took that legendary job at the Swiss patent office. It was during those years that he developed his special theory of relativity, in his spare time.

在我进一步谈论这个理论以及它如何引出矢量故事的下一步之前,让我先承认一下爱因斯坦与他的第一任妻子(她曾是他在理工大学的同学)的关系一直存在争议。

Before I talk more about this theory, and how it led to the next step in the story of vectors, let me acknowledge the ongoing controversy over Einstein’s relationship with his first wife, who’d been his fellow student at the Poly.

缅怀米列娃·马里奇

REMEMBERING MILEVA MARIĆ

关于这个悲剧故事,人们已经写了很多。在这个故事中,理想主义的年轻学生爱因斯坦和马里奇相爱了,他们秘密地生下(并放弃了)一个未婚婴儿,不顾父母的反对结婚,最终分居,爱因斯坦的名气越来越大,工作量越来越大,而马里奇屡次考试不及格,失去了信心和学术梦想,并承担起另外两个孩子的所有日常责任,这让他们分道扬镳。我不想再详细描述这个悲伤的故事了,我只想说,它不仅仅是他们两个人的问题。众所周知,爱因斯坦在两人关系破裂期间表现得非常糟糕,但两人关系破裂的种子是在他们还是学生时种下的,也与他们当时所处的父权文化有关。其中包括我认为两次让马里奇不及格的考官的性别歧视:她是班上唯一的女生,在最后一次考试时感到害怕并且怀孕了,但她之前的学校学习成绩优异。

Much has been written about this tragic saga, in which the idealistic young students Einstein and Marić fell in love, secretly had (and gave up) a baby out of wedlock, married against parental disapproval, and finally separated, torn apart by Einstein’s growing fame and workload—and the fact that Marić, having repeatedly failed her exams, lost her confidence and her academic dreams, and increasingly took on all the day-to-day responsibilities for their two other children. I won’t detail the sad drama further—except to say that it was bigger than the two of them. Einstein famously behaved very badly during the breakdown of the relationship, but the seeds of that breakdown, sown when they were still students, also have to do with the patriarchal culture in which they were both trapped. This includes what I believe is the sexism of the examiners who failed Marić twice: she was the only girl in her class, frightened and pregnant at the time of her final attempt at the exams, yet she had excelled in her earlier school studies.

至于人们普遍认为马里奇“为爱因斯坦做数学题”,或与爱因斯坦共同撰写了 1905 年的相对论论文,据我所知,没有确凿的证据证明这一点——她本人也从未声称过这一点。她无疑是第一批相信爱因斯坦的人之一,因为他与那些一直拒绝给他学术工作的“老庸人”作斗争——而她在理工学院的广泛学习至少使她成为爱因斯坦一个有价值且重要的倾听者。但他们之间现存的信件表明,她没有爱因斯坦那种驱动力强、富有创造力的科学好奇心。尽管如此,我还是忍不住想,如果他们今天上大学,社会更加开放,避孕措施也唾手可得,她就会拿到文凭和她计划中的博士学位,继续在科学上留下自己的印记——他们之间的关系可能会变得截然不同。10

As for the popular belief that Marić “did Einstein’s maths for him,” or coauthored his 1905 relativity paper, as far as I know there is no confirmed evidence for it—and she herself never claimed it. She was certainly one of the first to believe in Einstein, as he struggled against the “old philistines” who kept refusing him an academic job—and her extensive study at the Poly made her, at the very least, a worthy and important sounding board for him. But the surviving letters between them suggest she didn’t have Einstein’s driving, creative scientific curiosity. Still, I can’t help thinking that if they’d been at university today, with our more open society and with contraception readily available, she would have gotten her diploma and the doctorate she was planning, and gone on to make her mark in science—and things between them might have turned out very differently.10

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米列娃·马里奇-爱因斯坦和阿尔伯特·爱因斯坦于布拉格,1912 年。苏黎世联邦理工学院图书馆,Bildarchiv。摄影师:Jan F. Langhans/Portr_03106。公共领域。

Mileva Marić-Einstein and Albert Einstein in Prague, 1912. ETH-Bibliothek Zürich, Bildarchiv.Photographer: Jan F. Langhans/Portr_03106. Public domain.

狭义相对论

THE SPECIAL THEORY OF RELATIVITY

麦克斯韦的批评者抱怨他没有建立假设的以太模型,他们认为以太是光波传播的必要介质,就像声波需要空气一样。相反,他专注于描述具体、可测量的电磁效应——这是一个精明的举动,因为 1887 年,著名的迈克尔逊-莫雷实验“未能”通过最先进的干涉仪探测到以太。这个实验的概念实际上是麦克斯韦的,但将实验付诸实践所需的诺贝尔奖获奖设备是由阿尔伯特·迈克尔逊设计的。迈克尔逊是 1907 年第一位获得诺贝尔奖的美国人,他的实验为狭义相对论奠定了基础。因此,2017 年诺贝尔物理学奖授予了激光干涉引力波天文台 (LIGO) 的创始人,这具有很好的对称性,LIGO 在 2015 年探测到了引力波,正如广义相对论所预测的那样。

Maxwell’s critics had complained he had no model for the hypothetical ether, the medium they assumed must surely be necessary for the transmission of light waves, just as sound waves need air. Instead, he’d focussed on describing concrete, measurable electromagnetic effects—and it was an astute move, because in 1887, the famous Michelson-Morley experiment “failed” to detect the ether via a state-of-the-art interferometer. The concept for the experiment was actually Maxwell’s, but it was Albert Michelson who designed the Nobel Prize–winning equipment needed to put it into practice. Michelson was the first American to win a Nobel, in 1907, and his experiment helped set the stage for the special theory of relativity. So, there’s a nice symmetry in the fact that the 2017 Nobel Prize for physics went to the founders of the Laser Interferometer Gravitationalwave Observatory (LIGO), which detected gravitational waves in 2015, as predicted by the general theory of relativity.

1887 年实验背后的想法是,如果以太存在,当地球穿过它时,就会产生“以太风”——就像你在无风的日子骑自行车时感觉到脸上有微风一样。就像顺流而下比逆流或横流时速度更快,迈克尔逊和他的同事爱德华·莫雷预计光速在顺流方向最快——即“顺着”以太风。麦克斯韦曾提议利用木星卫星日食的时间,当时这颗巨行星在地球上几乎相反的两个轨道位置被观测到。然后地球在一个位置向木星移动,在另一个位置远离木星,这样来自木星系统的光速就可以同时“顺着”和“逆着”以太风来测量。11迈克尔逊和莫雷使用了干涉图样——与托马斯·杨最初用来显示光的波状性质的图样相同。具体来说,他们向顺流方向发送一束光,向横流方向发送另一束光——即平行于地球运动方向并垂直于地球运动方向;光速的任何差异都会导致两束光之间产生相位差,这将在干涉图样中显示出来。但研究人员没有发现这种差异。这意味着以太风对光速没有明显的影响。(从那时起,人们一直在进行实验,看看能否利用更好的设备发现微小的速度差异——一些物理学家谈到了“量子以太”的可能性。但以太的旧机械概念已经过时了。)

The idea behind the 1887 experiment was that if the ether existed, when Earth moved through it there would be an “ether wind”—just as you feel a breeze on your face when riding a bike on a still day. And just as you move faster swimming downstream in a river than upstream or across it, Michelson and his collaborator Edward Morley expected the speed of light would be fastest in the downstream direction—that is, “with” the ether wind. Maxwell had proposed using the timing of eclipses of Jupiter’s moons when the giant planet was seen from Earth at two nearly opposite positions in its orbit. Earth would then be moving toward Jupiter at one position and away from it at the other, so that the speed of the light from the Jovian system would be measured both “with” and “against” the ether wind.11 But Michelson and Morley used interference patterns—the same kind of patterns Thomas Young had used to show the wave-like nature of light in the first place. Specifically, they sent a beam of light downstream and another beam across-stream—that is, parallel to the direction of Earth’s motion and perpendicular to it; any difference in light speed would cause a phase difference between the two beams, which would show up in the interference pattern. But the researchers found no such difference. Which meant that the ether wind had no discernible effect on the speed of light. (Experiments have been ongoing ever since, to see if tiny speed differences can be found with better equipment—and some physicists speak of the possibility of a “quantum ether.” But the old mechanical idea of ether is out.)

到 1895 年,乔治·菲茨杰拉德和亨德里克·洛伦兹都各自巧妙地“解释”了迈克尔逊和莫雷的成果,他们认为物体(包括测量尺)在与以太风方向平行移动时会物理收缩。这种物理“长度收缩”会缩小光下游速度的测量值,掩盖光“实际上”更快的假设事实。

By 1895, both George FitzGerald and Hendrik Lorentz had independently and ingeniously “explained” Michelson and Morley’s result by suggesting that objects—including measuring rulers—physically shrank when they were traveling parallel to the direction of the ether wind. Such a physical “length contraction” would shrink the measurement of light’s downstream speed, masking the supposed fact that it was “actually” faster.

1904 年,洛伦兹将这一猜想应用于麦克斯韦电磁方程,开创了先河,设计了一组坐标变换,将静止观察者的参考系与相对于静止观察者以恒定速度运动的观察者的参考系联系起来,就像河中的游泳者相对于河岸上的观察者运动一样。这些变换旨在表明,运动参考系中的标尺以恰到好处的方式收缩,以解释迈克尔逊-莫雷结果——亨利·庞加莱将它们称为“洛伦兹变换”,当时他1905 年,爱因斯坦更全面地发展了洛伦兹的思想。(我们将在下面的图 9.3中看到这些转变。)多年后,在访问莱顿大学期间,爱因斯坦遇到了洛伦兹,并立即被他迷住了。他们发展了友谊,因为爱因斯坦认为洛伦兹既是一位异常清醒的思想家——他因在电子发现的背景下扩展了麦克斯韦理论而于 1902 年获得诺贝尔奖——也是一位品格完美的人。爱因斯坦称他为“我们这个时代最伟大、最高尚的人,智慧和精湛机智的奇迹。” 12

In 1904, Lorentz made a brilliant start at applying this conjecture to Maxwell’s equations of electromagnetism, devising a set of coordinate transformations between the frame of a stationary observer and that of an observer moving with constant speed relative to the stationary one, like the swimmer in the river moving relative to an observer on the riverbank. These transformations were designed to show that the rulers in the moving frame shrank in just the right way to account for the Michelson-Morley result— and Henri Poincaré dubbed them the “Lorentz transformations” when he developed Lorentz’s ideas more fully in 1905. (We’ll see these transformations in fig. 9.3 below.) Years later, during a visit to the University of Leiden, Einstein met Lorentz and immediately fell under his spell. They developed a friendship, for Einstein thought Lorentz was both an extraordinarily lucid thinker—he’d won a Nobel Prize in 1902 for his work extending Maxwell’s theory in light of the discovery of electrons—and a person of perfect character. “The greatest and noblest man of our times,” Einstein called him, “a marvel of intelligence and exquisite tact.”12

爱因斯坦和庞加莱从未对彼此产生好感,尽管他们彼此钦佩对方的工作。庞加莱是一位出色的数学家,同时也是一位开创性的科学哲学家。1905 年,他是巴黎索邦大学的著名数学天文学和天体力学教授,时年 51 岁。他 1905 年发表的关于洛伦兹变换的论文实际上是一个详细的相对论理论,在“特殊”情况下,观察者之间的相对运动(因此他们的坐标系之间的相对运动!)是恒定的。与此同时,26 岁的专利官爱因斯坦独立完成了他自己的“特殊”相对论。当马里奇读到他的最终稿时,她被迷住了,她告诉丈夫:“这是一部非常漂亮的作品!”一个多世纪后,物理学家仍然认为它很美:与庞加莱和洛伦兹论文的精彩但复杂的内容相比,爱因斯坦的论文更简单、更直观。它也是唯一一个抛弃了旧有的静止以太介质在“空”空间中传输光的概念的论文,因此它是唯一一个完全相对论的论文,我们将在接下来的几页中看到这一点。13

Einstein and Poincaré never warmed to each other, although they admired each other’s work. Poincaré was a superb mathematician—and a groundbreaking scientific philosopher to boot. In 1905 he was a famous fifty-one-year-old professor of mathematical astronomy and celestial mechanics at the Sorbonne in Paris. His 1905 paper on the Lorentz transformations was, in fact, a detailed theory of relativity, in the “special” case where the relative motion between observers (and therefore between their coordinate frames!) is constant. At the same time, twenty-six-year-old patent officer Einstein independently completed his own “special” theory of relativity. Marić had been captivated when she read his final draft, telling her husband, “It’s a very beautiful piece of work!” More than a century later, physicists still describe it as being beautiful: compared with the brilliant but convoluted complexities of Poincaré’s and Lorentz’s papers, Einstein’s is simpler and more intuitive. It is also the only one that ditched the old notion of a stationary ethereal medium for transmitting light through “empty” space, so it’s the only one that is fully relativistic, as we’ll see over the next couple of pages.13

由于庞加莱接受了以太存在的观点,并且洛伦兹变换提供了必要的“长度收缩”来支持以太假说,因此他以此为出发点。另一方面,爱因斯坦通过援引两个简单的原理开始分析相对运动。首先是相对论原理,该原理认为,如果一个静止的观察者推导出特定的物理定律,那么另一个以恒定速度相对于第一个观察者运动的观察者也应该推导出相同的定律。否则,物理学就没有什么意义了,如果每次改变速度时定律都会改变。庞加莱也理解这一点——至少在洛伦兹意义上,地球通过以太的运动对光没有影响,因此对麦克斯韦方程也没有影响。这意味着,正如洛伦兹、庞加莱和爱因斯坦所表明的那样,洛伦兹变换实际上就是正确的坐标变换,因此麦克斯韦方程具有相同的形式,无论电磁场矢量的分量是如何测量的——从固定坐标系,还是从相对于它以恒定速度运动的坐标系。

Since Poincaré accepted the idea that the ether existed—and that the Lorentz transformations provided the necessary “length contraction” to support the ether hypothesis—he’d taken this as his starting point. Einstein, on the other hand, began his analysis of relative motion by invoking two simple principles. First, the relativity principle, which says that if a stationary observer deduces a particular law of physics, then another observer, moving relative to the first one with constant speed, should deduce the same law. Otherwise, there wouldn’t be much point to physics, if the laws changed every time you changed your speed. Poincaré understood this, too—at least in the Lorentzian sense that Earth’s motion through the ether had no effect on light, and therefore no effect on Maxwell’s equations. What this means, as Lorentz, Poincaré, and Einstein all showed, is that the Lorentz transformations are, in fact, just the right coordinate transformations so that Maxwell’s equations have the same form, regardless of how the components of the electromagnetic field vectors are measured—from a fixed coordinate frame, or one that is moving relative to it with constant speed.

一个更简单的例子可以说明“保持相同形式”的含义,那就是牛顿第二定律。在图 9.3所示的简单伽利略坐标变换下,静止的观察者可以推导出水平力分量为F = mẍ,相对运动的观察者推导出F′ = mẍ′。该方程在每个帧中都有相同的形式,并且两个观察者都同意力等于质量乘以加速度。您可以在上面的框中看到对此的计算。(该框还显示了使用洛伦兹变换的计算,显示了狭义相对论中“长度收缩”的数学。我们的故事实际上不需要这些计算——我们只需要想法和结论——所以请随意浏览或跳过它们。)

A simpler example of what “keeping the same form” means is Newton’s second law. Under the simple, so-called Galilean coordinate transformation shown in figure 9.3, the stationary observer deduces the horizontal force component to be F = mẍ, and the relatively moving observer deduces F′ = mẍ′. The equation has the same form in each frame, and both observers agree that force equals mass times acceleration. You can see the calculations for this in the box above. (The box also shows calculations with the Lorentz transformations, showing the maths of “length contraction” in special relativity. We don’t really need these calculations for our story—we just need the ideas and conclusions—so feel free to skim or skip them.)

图像

图 9.3。两个坐标系SS′相对运动,S′相对S以恒定速度v向右移动。因此,我们假设S是站在人行道上的你,S′是开车经过的人。

FIGURE 9.3. Two frames S and S′ are moving relative to each other in such a way that S′ is moving to the right with constant speed v relative to S. So, let’s take S to be you standing on the sidewalk and S′ to be someone driving by.

如果轴线最初重合(即在时间t = 0时你和汽车处于同一水平),那么在时间t之后, S 参考系(汽车)将向右移动vt个单位。在这种情况下,相对运动是水平的,因此对于任何点 P, yy′坐标(在三维空间中,zz′坐标也是如此)将相同;然而,在时间t期间, S′已移近P ,因此在此时进行的测量将使P的水平坐标在S参考系中为x,而在相对于移动的S′参考系测量时为x′(= xvt ) 。

If the axes initially coincided—if you and the car were level with each other at time t = 0—then after a time t the S′ frame (the car) will have moved vt units to the right. In this case the relative motion is horizontal, so for any point P the y and y′ coordinates (and in 3-D the z and z′ coordinates too) will be the same; during the time t, however, S′ has moved closer to P, so a measurement made at that moment will give P’s horizontal coordinate as x when measured in the S frame and x′ (= xvt) when measured with respect to the moving S′ frame.

这是牛顿/伽利略的视角,时间对于两个观察者来说以相同的速率流逝。即便如此,你还是可以看到,在不同的相对运动的参考系中,距离的测量结果必须不同,因此速度的测量结果也必须不同。你可以在相关的(完全可选的!)框中从牛顿和爱因斯坦的角度看到这种现象的数学后果。

This is the Newtonian/Galilean perspective, where time ticks away at the same rate for both observers. Even so, you can see that measurements of distance must be different in the different, relatively moving frames, and therefore speeds must be measured differently, too. You can see the mathematical consequences of this, from both the Newtonian and Einsteinian perspectives, in the related (and entirely optional!) box.

坐标变换和不变性

COORDINATE TRANSFORMATIONS AND INVARIANCE

图 9.3 的计算

CALCULATIONS FOR FIGURE 9.3

图 9.3中水平伽利略平移的坐标变换为:

The coordinate transformations for the horizontal Galilean translation in figure 9.3 are:

x′ = xvt,y′ = y,z′ = z,t′ = t(1)

x′ = xvt, y′ = y, z′ = z, t′ = t. (1)

当v远小于光速时,这种方法很有效。在这种“日常”情况下,使用(1)你会发现,无论以S还是S′为单位,牛顿定律的形式都保持不变:比如,如果你推导出F = mẍ,那么行驶中的汽车中的人就会推导出F′ = mẍ′。这是因为x′ = xvt ,当你对t′(= t)对其进行两次微分时,你会得到(使用牛顿的点符号表示时间导数);但由于速度是恒定的,第二项为零,因此有ẍ′ = 。所以,应用于这个力的分量的牛顿第二定律在两个框架中都有相同的形式F = mẍ = mẍ′ = F′。事实上,力的分量在每个框架中都有相同的值,所以它们在伽利略变换下是不变的;并且由于yz方向上没有相对运动,因此两个观察者也将在这些方向上测量相同的力分量。

It works well when v is much less than the speed of light. In such “everyday” situations, using (1) you find that Newton’s laws keep their form, regardless of whether they are measured in S or S′: for instance, if you deduce F = mẍ, then the person in the moving car deduces F′ = mẍ′. That’s because x′ = xvt, and when you differentiate this twice with respect to t′ (= t), you get (using Newton’s dot notation for the time derivatives); but since the speed is constant, the second term is zero and you have ẍ′ = . So Newton’s second law applied to this component of force has the same form in both frames: F = mẍ = mẍ′ = F′. In fact, the force components have the same value in each frame, so they are invariant under Galilean transformations; and since there’s no relative motion in the y and z directions, both observers will measure the same force components in those directions, too.

尽管你们都推导出相同的 F 和 ẍ 值,但你们对运动物体(例如被抛出的球)的水平速度分量u存在分歧,因为在S 中,u = ,而在S′ 中,u′ = ẋ′ = v。因此,力和加速度在伽利略变换下不变,但速度不是。

Although you both deduce the same values of F and ẍ, you disagree on the horizontal speed component u of a moving object such as a ball being thrown, because in S, u = , and in S′, u′ = ẋ′ = v. So, the force and acceleration are invariant under Galilean transformations, but the speed is not.

麦克斯韦方程在坐标变换(1)下不保持其形式。相反,洛伦兹变换才是电磁定律的正确变换。(在这种情况下,各个分量以及电场和磁场矢量实际上都不是不变的:例如,如果电荷在一个框架中处于静止状态,相对运动的观察者将看到磁场,但在电荷框架中的观察者则不会。但连接矢量的麦克斯韦方程在每个框架中都有相同的形式。)这些变换表明,不仅空间测量而且时间测量都是相对的——因此你不能再假设t′ = t。图中设置的洛伦兹变换为:

Maxwell’s equations do not keep their form under the coordinate transformations (1). Instead, the Lorentz transformations are the right ones for the laws of electromagnetism. (In this case, the individual components and indeed the electric and magnetic field vectors are not invariant: e.g., if a charge is at rest in one frame, a relatively moving observer will see a magnetic field, but an observer in the charge’s frame will not. But the Maxwell equations linking the vectors do have the same form in each frame.) These transformations show that not just spatial measurements but also time measurements are relative—so you can no longer assume that t′ = t. The Lorentz transformations for the set-up in the diagram are:

=β=c2 (2)

x=βxvt,y=y,z,ttvxc2, (2)

其中c是光速(测量单位通常选择为c = 1),并且

where c is the speed of light (measurement units are often chosen so that c = 1), and

β=1/12/c2 (3)

β=1/1v2/c2. (3)

这些方程表示x方向上所谓的增强(S′相对于S),但完整的洛伦兹变换可以描述任何方向的增强,以及旋转。请注意,如果c是无限的,正如超距作用所暗示的那样,那么方程(2)就是方程(1)

These equations represent a so-called boost (of S′ relative to S) in the x-direction, but the full Lorentz transformations can describe boosts in any direction, and rotations, too. And note that if c is infinite as implied in action-at-a-distance, then equations (2) are just equations (1).

如果你要测量一辆行驶中的汽车的长度,在给定的时间,你必须同时测量它的两个端点x ax b,并计算x bx a;使用洛伦兹变换(2)将你的结果与汽车在其自身框架中的实际(“静止”)长度x bx a进行比较,你会得到

If you were to measure the length of a moving car, at a given time you would have to measure simultaneously both its endpoints xa and xb, say, and compute xbxa; using the Lorentz transformations (2) to compare your result with the actual (“rest”) length of the car in its own frame, xbxa, you get

xb′xa = β ( xb xa )

xbxa = β(xbxa)

(因为对于同时测量,t bt a = 0)。由于 β > 1(因为对于非零v ,分母小于 1 ),您会发现实际的汽车长度大于您测量的长度。换句话说,您的测量表明汽车已经“缩小”。与洛伦兹的想法不同,汽车本身并没有物理、分子上的缩小,而是计算中使用的测量值缩小了。但这些测量值确实对静止观察者的物理学产生了真实的、可测试的结果。

(because tbta = 0 for a simultaneous measurement). Since β > 1 (because the denominator is less than 1 for nonzero v), you see that the actual car length is greater than the one you measured. In other words, your measurement suggests the car had “shrunk.” Unlike Lorentz’s idea, there is no physical, molecular shrinking of the car itself, but a shrinking in measurements used in calculations. But these measurements do have real, testable consequences in the stationary observer’s physics.

类似地,时间测量的相对性也是地球上的观察者推断时间对于快速移动的飞机或宇宙飞船旅行者(或 GPS 卫星)来说会变慢的原因,就像距离缩短一样。(正如我们将看到的,狭义相对论和广义相对论都是解释 GPS 测量中的时间的必要条件。)

Similarly, the relativity of the time measurement is why Earth-based observers deduce that time slows down for fast-moving airplane or spaceship travelers—or GPS satellites—just as distances shrink. (Both special and general relativity are needed to account for time in GPS measurements, as we’ll see.)

然而,正如爱因斯坦所意识到的,你也可以将图 9.3解释为S相对于S′移动——因此它向左移动,其速度为−v。因此,从这个角度来看,洛伦兹变换为:

As Einstein realised, however, you can also interpret figure 9.3 as saying that S is moving relative to S′—so it is moving to the left, and its speed is −v. So, the Lorentz transformations from this point of view are:

=β+===β+c2 (4)

x=βx+vt,y=y,z=z,t=βt+vxc2. (4)

而这一次,测量的是S框架内收缩的长度。

And this time, it is measured lengths in S’s frame that contract.

团体

GROUPS

群是研究对称性(如不变性)的重要工具。在这种情况下,群的“元素”是坐标变换,如果它们都遵循一组简单的规则,它们就会形成一个群,我将以洛伦兹变换为例进行说明(尽管在这个故事中我们不需要这些细节)。

Groups are important tools for studying symmetries, such as invariance. In this case, the “elements” of the group are coordinate transformations, and they form a group if they all obey a simple set of rules, which I’ll illustrate for the Lorentz transformations (although we won’t need these details in this story).

洛伦兹变换(2)有一个“逆” (4),还有一个“恒等”元素(即,变换不会改变的元素——在本例中为v = 0 时的变换),这一事实是了解它们形成数学“群”的关键。封闭性(群中所有可能的变换都属于同一类型的思想)和结合性(群“乘积”——在本例中为两个变换的复合)是其他关键特征。

The fact that the Lorentz transformations (2) have an “inverse,” (4), and also an “identity” element (that is, an element that is unchanged by the transformation—in this case, the transformation when v = 0), is key to knowing that they form a mathematical “group.” Closure (the idea that all possible transformations in the group are of the same type) and associativity (of the group “product”—in this case, the composition of two transformations) are the other key features.

类似地,如果你推断×=我们之前看到的麦克斯韦方程之一,那么使用洛伦兹变换,你会看到移动的观察者推导出一个具有完全相同形式的方程。所以,你们都同意方程右边的变化磁场会引起左边电磁场的旋度。其他全矢量麦克斯韦方程也是如此。

Similarly, if you deduce ×E=Bt, one of Maxwell’s equations that we saw earlier, then using the Lorentz transformations you’d see that a moving observer deduces an equation with exactly the same form. So, you’d both agree that the changing magnetic field on the right-hand side of the equation gives rise to the curl of the electromagnetic field on the left. The same goes for the other whole-vector Maxwell equations.

换句话说,如果表示物理定律的方程即使在不同的坐标系中测量其分量也具有相同的形式,那么所有观察者都会得出相同的物理结论(以及相同的数学结论,正如我们在不变方程ab = 0 中看到的那样)。这就是相对论原理在起作用。

In other words, if an equation representing a law of physics has the same form even when its components are measured in different frames, then all observers will deduce the same physics (and the same maths, as we saw with the invariant equation ab = 0). That’s the principle of relativity in action.

这也是不变性的另一个例子——在这种情况下,方程形式的不变性。(这种形式不变的方程也称为“协变”方程。)14

It’s also another example of invariance—in this case, the invariance of the form of the equations. (Such form-invariant equations are also called “covariant.”)14

爱因斯坦的第二条原理是真空中的光速与光源的运动无关(物理学家似乎通过实验发现了这一点)。这两个原理结合起来意味着真空光速c是一个通用常数。

Einstein’s second principle was that the speed of light in empty space is independent of the motion of the source (which is what physicists seemed to have found experimentally). Together these two principles imply that the vacuum speed of light, c, is a universal constant.

仅凭这两个原理,爱因斯坦就从头推导出了洛伦兹变换,而且是完全一般性的。相比之下,庞加莱则认为洛伦兹的“长度收缩”是一种真实的物理效应,然后通过证明洛伦兹变换保持麦克斯韦方程形式不变,重新推导出洛伦兹变换。(沃尔德马尔·福格特早在 1887 年就发现了类似的东西,所以这些想法还“悬而未决”。我们将在第 11 章中听到更多关于福格特的内容。)但爱因斯坦不需要关于以太的假设,也不需要假设标尺有任何客观的物理收缩——相反,你会从不同的、相对移动的参考系得到不同的测量结果。我在图 9.3 的计算框中展示了一个简单的例子,但关键的事实,正如爱因斯坦所表明的,是这种效应是相互的:每个观察者都可以认为自己处于静止状态,而另一个观察者则认为在运动。换句话说,每个观察者都会测量到对方尺子的收缩,因为他们彼此相对运动。相反,洛伦兹和庞加莱认为只有一个观察者“真正”在运动(相对于无所不在的以太而言)——即尺子“真正”收缩的观察者。这就是为什么爱因斯坦的理论是唯一完全相对论的理论。

From these two principles alone, Einstein had derived the Lorentz transformations from scratch, in a completely general way. By contrast, Poincaré had assumed Lorentz’s “length contraction” was a real physical effect, and then rederived the Lorentz transformations by showing that they left the form of Maxwell’s equations invariant. (Woldemar Voigt found something similar back in 1887, so these ideas were “in the air.” We’ll hear more about Voigt in chap. 11.) But Einstein had no need of hypotheses about the ether and no need to suppose any objective physical shrinking of rulers took place—rather, you get different measurements from different, relatively moving frames of reference. I showed a simple example of this in the box of calculations for figure 9.3, but the key fact, as Einstein made clear, is that the effect is reciprocal: each observer can consider themselves at rest and the other one to be moving. In other words, each observer would measure the other’s ruler contracting, because they’re each moving relative to the other. Lorentz and Poincaré, by contrast, believed that only one observer was “really” moving (relative to the all-pervading ether)— the one whose ruler “really” shrinks. That’s why Einstein’s was the only fully relativistic theory.

四维时空需要四维矢量分析

FOUR-DIMENSIONAL SPACE-TIME NEEDS FOUR-DIMENSIONAL VECTOR ANALYSIS

洛伦兹变换包括时间以及三个空间方向(如上图9.3中方程(2)所示)。因此,当如今提到“相对论”一词,很多人首先想到的便是时空的四维本质。然而“第四维”的概念自 19 世纪 80 年代起就一直吸引着公众。诚然,汉密尔顿在 1843 年定义四元数时创造了一个数学上的四维空间——但正如我们在矢量之争中看到的那样,甚至数学家们也无法就它的价值达成一致。因此,当泰特、凯莱、亥维赛等人在 19 世纪 80 年代和 90 年代争论分量、整体矢量与四元数的学术价值时,字面意义上的四维空间的可能性正在一些流行书籍中得到探讨——例如数学家查尔斯·霍华德·辛顿的《科学浪漫史》《思想的新时代》

The Lorentz transformations include time as well as the three spatial directions (as you can see in equation (2) in the box above with fig. 9.3). So, when the term “relativity” is mentioned today, one of the first things many people think of is the four-dimensional nature of space-time. Yet the idea of a “fourth dimension” had been tantalising the public since the 1880s. True, Hamilton had created a mathematical four-dimensional space when he defined quaternions in 1843—but as we saw with the vector wars, not even mathematicians could agree on its worth. So, while Tait, Cayley, Heaviside, and the others debated the academic merit of components versus whole vectors versus quaternions in the 1880s and 1890s, the possibility of a literal four-dimensional space was being tackled in popular books—such as mathematician Charles Howard Hinton’s Scientific Romances and A New Era of Thought.

辛顿(George Boole 的女婿)也是一位唯灵论者,唯灵论者喜欢神秘的另一个维度的想法。这些想法已经渗透到大众文化中,以至于当 Grace Chisholm 参加与她的博士论文(包括n维空间)相关的口试时,她的一位考官提到了精神观念,并问所说的“更高维度”是什么意思。她回答说,对她来说,这只是谈论数学中某些抽象关系的一种方式。她以最高荣誉获得了博士学位,尽管显然她回答过比这难得多的问题!15

Hinton—who was George Boole’s son-in-law—was also a spiritualist, and spiritualists loved the idea of a mysterious other dimension. These ideas had so penetrated popular culture that when Grace Chisholm stood for the oral exam related to her doctoral dissertation (which included n-dimensional spaces), one of her examiners mentioned the spirit idea, and asked what she meant by “higher dimensions.” She replied that for her, it was simply a way of speaking about certain abstract relations in mathematics. She got her doctorate with highest honours, although obviously she’d fielded much harder questions than this!15

至于辛顿,他并没有完全沉迷于鬼神,他还专注于尝试将四维几何物体形象化,比如“超立方体”。你可以将立方体想象成“超正方形”,即一串二维正方形,它们组成一个三维形状——也就是一个立方体——所以超立方体就是四维空间中的立方体排列。1954 年,萨尔瓦多达利在他的耶稣受难画作《Corpus Hypercubus》中借鉴了这一想法,其中的十字架是由立方体组成的——但在 70 年前,辛顿曾启发校长埃德温阿博特在 1884 年创作了传奇的《平面国:多维罗曼史》。辛顿还启发了他妻子的妹妹艾丽西亚布尔斯托特,也就是布尔最小的女儿。斯托特在四维几何中发现了几个令人费解的结果,其中包括她用六百四面体构成的想象中的四维结构的各个部分制作的一组物理模型,可以将它们合在一起看作是二十面体的四维类似物的三维表面。这种视觉几何想象让人难以置信。16

As for Hinton, he hadn’t lost his mind completely to the spirits, for he also focussed on trying to visualise four-dimensional geometric objects, such as “hypercubes.” You can think of a cube as a “hypersquare,” a sequence of two-dimensional squares that forms a 3-D shape—that is, a cube—so a hypercube would be an arrangement of cubes in 4-D space. In 1954, Salvador Dali famously drew on this idea in his crucifixion painting “Corpus Hypercubus,” in which the cross is made of cubes—but seventy years earlier, Hinton had inspired the schoolmaster Edwin Abbott to write his legendary 1884 Flatland: A Romance of Many Dimensions. Hinton also inspired his wife’s sister Alicia Boole Stott, Boole’s youngest daughter. Stott discovered several mind-bending results in four-dimensional geometry, including a collection of physical models she made of various sections of an imagined 4-D structure made of six hundred tetrahedrons, which together can be thought of as the three-dimensional surface of the four-dimensional analogue of an icosahedron. The mind boggles at such visual geometric imagination.16

随后,在 1895 年——就在洛伦兹发现洛伦兹变换来解释迈克尔逊-莫雷实验的同一年——赫伯特·乔治·威尔斯 (HG Wells) 出版了他著名的小说《时间机器》,他在书中声称时间是第四维。然而,尽管威尔斯的想象力极其丰富,但仅靠言语还不足以让这个想法持久存在——使其成为现实。后来洛伦兹、庞加莱和爱因斯坦才揭示出这种四维结构的数学性质——而爱因斯坦则通过提出可检验的预测来检验他的理论,使它成为“现实”。(其中一个预测让他完全出乎意料地推断出E = mc2 )所以,今天当人们听到“第四维”这个词时,他们往往会想到爱因斯坦,而不是威尔斯——爱因斯坦用数学语言做到了这一点。

Then, in 1895—the same year that Lorentz found the Lorentz transformations to explain the Michelson-Morley experiment—H. G. Wells published his famous novel The Time Machine, in which he claimed that time was the fourth dimension. Brilliantly imaginative as Wells was, however, words alone were not enough to make this idea stick—to make it real. It took Lorentz, Poincaré, and Einstein to uncover the mathematical properties of such a four-dimensional construct—and it took Einstein to make it “real” by suggesting testable predictions to check his theory. (One of those predictions led him, completely unexpectedly, to deduce that E = mc2.) So, when people hear the words “fourth dimension” today, they tend to think of Einstein, not Wells—and Einstein did it with the language of maths.

向量语言而言,庞加莱完全以分量的形式工作,但洛伦兹也使用全向量符号和向量微积分。爱因斯坦和庞加莱一样,在他的狭义相对论论文中以分量的形式写下所有方程,尽管他的坐标(t、x、y、z)既代表时间,又代表空间,但他还没有提到时空。

As far as vector language is concerned, Poincaré worked entirely in components, but Lorentz also used whole-vector notation and vector calculus. In his special relativity paper Einstein, like Poincaré, wrote all his equations in terms of components, and although his coordinates (t, x, y, z) represented both time and space, he did not yet speak of space-time.

原则上,爱因斯坦可以使用四元数(有四个分量)来表示四维坐标系中的数量。17不过,他可能从未听说过四元数——他的数学教授闵可夫斯基曾说过,英国以外的人都不会使用四元数——也许他甚至不太了解向量代数,尽管闵可夫斯基确实教过这门课。在学生时代,爱因斯坦认为,如果要成功解开自然界的秘密,他需要一个单一的焦点,而这个焦点就是物理学。此外,他觉得数学中有太多主题可供选择,让他不知所措。因此,尽管他钦佩闵可夫斯基,但他还是逃了不少课,以便自学课堂上没有教的物理学——包括麦克斯韦理论。另一方面,闵可夫斯基认为爱因斯坦只是“一只懒狗,根本不关心数学”。相比之下,当闵可夫斯基还是一名 17 岁的学生时,他就因他的数学工作——并偷偷把奖金给了一位贫困的同学。18

In principle Einstein could have used quaternions, which have four components, to represent quantities in his four-dimensional coordinate system.17 He’d likely never heard of quaternions, though—his maths professor, Minkowski, had said no one outside Britain used them—and perhaps he didn’t even know much vector algebra, although Minkowski did teach that. As a student Einstein had figured that he needed a single focus if he was to succeed in unraveling nature’s secrets, and this focus was firmly on physics. Besides, he felt there were so many topics to choose from in maths that he was overwhelmed. So, although he admired Minkowski, he’d cut quite a few of his classes so he could study on his own the physics he wasn’t being taught in lectures—including Maxwell’s theory. Minkowski, on the other hand, thought Einstein was simply “a lazy dog who never bothered about mathematics at all.” By contrast, when Minkowski himself had been a seventeen-year-old student, he’d won a prestigious award for his mathematical work—and had secretly given the prize money to an impoverished classmate.18

1905 年,闵可夫斯基当然食言了:“爱因斯坦总是缺课——我真的不相信他能做到!”但尽管爱因斯坦的论文概念优雅,但与庞加莱的论文相比,从数学上讲,年轻的爱因斯坦的论文还是有些粗糙。事实上,现任德国著名大学哥廷根大学纯数学教授的闵可夫斯基告诉他的学生:“爱因斯坦对他的深奥理论的陈述在数学上很笨拙——我可以这么说,因为他在苏黎世接受了我的数学教育。” 19

Minkowski certainly ate his words in 1905: “Oh that Einstein, always missing lectures—I really would not have believed him capable of it!” But for all its conceptual elegance, compared with Poincaré’s paper young Einstein’s was a little rough around the edges, mathematically speaking. In fact, Minkowski, now professor of pure mathematics at Germany’s prestigious Göttingen University, told his students, “Einstein’s presentation of his deep theory is mathematically awkward—I can say that because he got his mathematical education in Zurich from me.”19

例如,庞加莱精通不变性和群论的数学语言,并用它来展示一些令人惊讶的东西:四维表达式

For instance, Poincaré was fluent in the mathematical language of invariance and group theory, and used it to show something surprising: the four-dimensional expression

x 2 + y 2 + z 2 − ( ct ) 2

x2 + y2 + z2 − (ct)2

如果通过洛伦兹变换变换坐标,它不会改变。换句话说,它在该变换“群”下不变。(实际上,庞加莱在这里使用了c = 1,今天在洛伦兹变换和其他相对论方程中,选择单位使得c = 1 是很常见的。)爱因斯坦也发现了同样的结果,尽管他并没有用如此复杂的方式表达它。正式的“群论”是由埃瓦里斯特·伽罗瓦开创的——他是一位冲动的革命煽动者和鲁莽的情人,他在 1832 年的一场决斗中死去,当时他年仅 20 岁。凯莱是众多对群论做出重要早期贡献的人之一,我将其主要特点列在了图 9.3 的方框标题末尾。但真正理解这种不寻常表达方式的人是闵可夫斯基。

doesn’t change if you transform the coordinates via Lorentz transformations. In other words, it is invariant under this “group” of transformations. (Actually, Poincaré used c = 1 here, and today it is common to choose units so that c = 1 in the Lorentz transformations and other equations of relativity.) Einstein found the same result, although he didn’t express it in such a sophisticated way. Formal “group theory” was pioneered by Évariste Galois—the impetuous prorevolutionary agitator and foolhardy lover who famously died in a duel in 1832, when he was just twenty years old. Cayley was one of the many others who made important early contributions to group theory, whose key features I listed at the end of the boxed caption to figure 9.3. But it was Minkowski who really made sense of this unusual expression.

这很不寻常,因为正如庞加莱所指出的那样,物理学家习惯于所有项都相加的二次表达式。例如,位置向量的长度可以通过毕达哥拉斯定理从其分量中找到(通过将图 0.2扩展到 3-D 并简单地用坐标表示分量,您可以看到):

It was unusual because, as Poincaré had pointed out, physicists were used to quadratic expressions where all the terms were added. For example, the length of a position vector is found from its components via Pythagoras’s theorem (as you can see by extending fig. 0.2 to 3-D and representing the components simply in terms of the coordinates):

一个=++一个=2+2+2

a=xi+yj+zka=x2+y2+z2.

图像

赫尔曼·闵可夫斯基,约。 1896年(当时爱因斯坦是他的学生)。苏黎世联邦理工学院图书馆,Bildarchiv/摄影师未知/Portr_02711。公共领域。

Hermann Minkowski, ca. 1896 (when Einstein was his student). ETH-Bibliothek Zürich, Bildarchiv/Photographer unknown/Portr_02711. Public domain.

这是在平坦的三维欧氏空间中测量长度和距离的公式,它在坐标变换(例如图 9.1中的旋转)下不变。闵可夫斯基意识到,狭义相对论中的类似概念是以下表达式:

This is the formula for measuring lengths and distances in flat, 3-D Euclidean space, and it is invariant under coordinate transformations such as the rotations in figure 9.1. Minkowski realised that the analogous concept in special relativity is the expression:

2+2+2c2

x2+y2+z2ct2,

庞加莱和爱因斯坦已经证明了,在洛伦兹变换下不变。这个表达式看起来很像欧几里得距离公式,但它不是普通空间中的距离度量。相反,它是闵可夫斯基所说的“时空”中的“距离”。换句话说,它是时空中“事件”之间的间隔,而不是空间中点之间的距离。你可以在尾注中看到它的物理含义,但在这里我关注的是它与欧几里得距离公式的数学类比。20

which Poincaré and Einstein had shown is invariant under Lorentz transformations. This expression looks rather like the Euclidean distance formula, but it is not the measure of distance in ordinary space. Rather, it is the “distance” in what Minkowski called “space-time.” In other words, it is the interval between “events” in space-time, rather than the distance between points in space. You can see what it means physically in the endnote, but here I’m focusing on its mathematical analogies with the Euclidean distance formula.20

狭义相对论中使用的时空现在被称为“闵可夫斯基时空”;它是平坦欧几里得空间的四维延伸,因此也被称为“平坦”时空,与弯曲的欧几里得时空相对。广义相对论。距离或间隔度量被称为“度量”,形式为2+2+2c2为了纪念他,我们把这个度量称为闵可夫斯基度量。(我们稍后会看到更精确的定义。)“世界线”的概念也是闵可夫斯基提出的。在欧几里得几何中,物体位于空间中的某个;但即使物体在空间中静止不动,它也在时间中移动,因此它在时空中的位置由一条通过该点的线表示——这条线与时间轴平行,并且随着每一秒的流逝而变长。

The space-time used in special relativity is now called “Minkowski space-time”; it is a four-dimensional extension of flat Euclidean space, so it is also called “flat” space-time, as opposed to the curved space-times of general relativity. A distance or interval measure is called a “metric,” and a measure with the form x2+y2+z2ct2 is called the Minkowski metric in his honour. (We’ll see a more precise definition later.) The concept of “world lines” is due to Minkowski, too. In Euclidean geometry, objects are located at a point in space; but even if the object is stationary in space, it is moving in time, so its location in space-time is represented by a line through the point—a line that is parallel to the time axis, and which gets longer as each second ticks by.

1907 年 11 月,闵可夫斯基在一次演讲中首次提出了新的时空概念。一年后,他在著名的演讲《空间与时间》中更充分地阐述了这一概念。演讲以令人难忘的宣言开场:“从今以后,空间本身和时间本身注定要消失,化为虚影,只有两者的某种结合才能保持独立的现实。”他赞扬爱因斯坦认识到了真正的双向相对论原理,礼貌地驳斥了洛伦兹关于运动物体的字面物理收缩的“幻想”假设。这次演讲的自信语气——至少在纸面上——掩盖了温和的闵可夫斯基过去在观众面前脸红结巴的事实。21

Minkowski first put forward his new concept of space-time in a lecture he gave in November 1907. A year later, he developed this more fully in his famous talk, “Space and Time,” which opened with the memorable proclamation, “Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.” He gave Einstein the credit for recognising the true, two-way principle of relativity, politely dismissing Lorentz’s “fantastical” hypothesis of a literal, physical contraction of moving objects. The confident tone of this address—on paper, at least—belies the fact that the gentle Minkowski used to turn deep red and stammer in front of an audience.21

不过,我们故事的关键在于,闵可夫斯基随后开始发展四维向量分析。有一次,他甚至尝试了四元数,因为正如我们之前所见,你可以用四元数除法,但不能用向量除法。最后,他发现普通的海维赛德-吉布斯式向量分析更加灵活。

The key thing for our story, though, is that Minkowski then made a start at developing four-dimensional vector analysis. At one point he even tried quaternions, because you can divide with quaternions but not vectors, as we saw earlier. In the end, he found the ordinary, Heaviside-Gibbs–style vector analysis more flexible.

在普通的向量分析中,向量分析发生在图 9.1所示的平坦欧氏空间中,向量的长度也可以用它与自身的标量积来表示:

In ordinary vector analysis, which takes place in the flat Euclidean space used in diagrams such as figure 9.1, the length of a vector can also be written in terms of its scalar product with itself:

一个=++一个=一个一个=2+2+2

a=xi+yj+zka=aa=x2+y2+z2.

在(平坦)时空中,闵可夫斯基用区间度量通过类比定义了标量积:如果 4 维向量a具有分量 ( x, y, z, t ),则标量积为

In (flat) space-time, Minkowski defined the scalar product by analogy, using the interval metric: if a 4-D vector a has components (x, y, z, t), then the scalar product is

a a = x2 + y2 + z2− ct 2

aa = x2 + y2 + z2 − (ct)2.

更一般地,使用现代符号(并选择单位使得c = 1),而两个三维向量的标量(或点积)是

More generally, and using modern notation (and choosing units so that c = 1), while the scalar (or dot) product of two 3-D vectors is

a b = a 1 b 1 + a 2 b 2 + a 3 b 3

ab = a1b1 + a2b2 + a3b3,

闵可夫斯基时空中的四维点积是

the 4-D dot product in Minkowski space-time is

a b = a 1 b 1 + a 2 b 2 + a 3 b 3 a 4 b 4

ab = a1b1 + a2b2 + a3b3a4b4.

(今天,用后缀 0 而不是 4 来表示时间分量也很常见。)后来的数学家将其推广到任何类型的空间——弯曲或平坦、3 维、4 维或n维——通过度量定义标量积,这是一个体现数学类比力量的美丽例子。

(Today, it’s also common to denote the time component with a suffix 0 instead of 4.) Later mathematicians will generalise this to any kind of space—curved or flat, 3-D, 4-D, or n-D—defining the scalar product via the metric, in a beautiful example of the power of mathematical analogies.

在另一篇论文中,闵可夫斯基不仅接近 4 维矢量分析,还涉足张量分析。不幸的是,他没有机会充分发展他的工作,也没有机会知道今天他的名字仍存在于“闵可夫斯基度量”中。在他为世界贡献时空后不久,他因阑尾炎手术后突然去世。他只有四十四岁。当他的老同学、哥廷根同事大卫·希尔伯特站在学生面前告诉他们这个不幸的消息时,他哭了。22

In another paper, Minkowski came close not just to 4-D vector analysis; he dipped his toe into tensor analysis, too. Tragically, he never got the chance to develop his work fully, or to know that today his name lives on in the “Minkowski metric.” Not long after he gave space-time to the world, he died suddenly, after surgery for appendicitis. He was only forty-four. When his old college friend and Göttingen colleague, David Hilbert, stood in front of his students to tell them the sad news, he wept.22

• • •

• • •

起初,爱因斯坦对他的前教授对他的理论所做的贡献并不满意——在那个阶段,他更喜欢坐标和分量,而不是整个无坐标的矢量。因此,闵可夫斯基的朋友阿诺德·索末菲继承了他巧妙的时空发明,开始探索标量和矢量积的四维类似物,以及散度、旋度和梯度的矢量微积分运算。索末菲当时是慕尼黑大学的理论物理学教授,他在 1910 年的论文《论相对论 I:四维矢量代数》的开头向“突然去世的朋友”闵可夫斯基致敬。

At first, Einstein wasn’t impressed with what his former professor had done to his theory—at that stage he preferred coordinates and components to whole, coordinate-free vectors. So, it was Minkowski’s friend Arnold Sommerfeld who took up his ingenious space-time invention and began to explore 4-D analogues of scalar and vector products, and of the vector calculus operations of divergence, curl, and grad. Sommerfeld, who was then a professor of theoretical physics at the University of Munich, opened his 1910 paper, “On the Theory of Relativity I: Four-Dimensional Vector Algebra,” with a tribute to Minkowski, “the friend who suddenly passed away.”

索末菲曾是 1903 年成立的德国“矢量委员会”的成员。在英国矢量战争之后,该委员会的目标是确定一个标准的矢量符号——尽管它的创始人、哥廷根数学教授菲利克斯·克莱因 (Felix Klein) 认为,它所取得的成就只是符号的激增!克莱因是赫维赛德 (和麦克斯韦) 的粉丝,但格拉斯曼的《引力理论》也给他留下了深刻的印象。格拉斯曼的儿子贾斯特斯于 1869 年进入哥廷根大学学习数学后不久,克莱因就听说了这本书——贾斯特斯自豪地带来了他父亲的书的副本,专门留给两位对格拉斯曼的工作感兴趣的教授。其中一位教授是阿尔弗雷德·克莱布施 (Alfred Clebsch),他对格拉斯曼的热情也激励了他的同事克莱因。二十​​年后,克莱因和吉布斯一起促成了格拉斯曼文集的出版。23与此同时,索末菲先是成为克莱因在哥廷根的助手,然后成为他的合作者。 1906 年,他移居慕尼黑,并把克莱因培养的对向量和不变量的兴趣带到了这里。

Sommerfeld had been a member of Germany’s “Vector Commission,” which was established in 1903. In the wake of Britain’s vector wars, the commission’s goal was to settle on a standard vector notation—although its founder, Göttingen maths professor Felix Klein, thought that all it achieved was a proliferation of notations! Klein was a fan of Heaviside (and Maxwell), but he had also been impressed with Grassmann’s Ausdehnungslehre. He’d heard about it soon after Grassmann’s son Justus enrolled as a maths student at Göttingen in 1869—Justus had proudly brought along copies of his father’s book, earmarked for two professors who had expressed interest in Grassmann’s work. One of these professors was Alfred Clebsch, whose enthusiasm for Grassmann also inspired his colleague Klein. Two decades later, Klein—along with Gibbs—was instrumental in the publication of Grassmann’s collected works.23 At the same time, Sommerfeld had become first Klein’s assistant at Göttingen and then his collaborator. He moved to Munich in 1906 and had taken with him the interest in vectors— and invariants—that Klein had nurtured.

在 1910 年发表的两篇关于相对论的论文中,索末菲引入了“四矢量”一词来表示时空中的四维矢量。他强调了时空间隔不变性的重要性,因为它表达了光速的恒定性(如尾注24所示)。庞加莱、洛伦兹和爱因斯坦当然也知道这一点。但索末菲还指出了不变符号在将麦克斯韦方程应用于时空时的重要性,并指出洛伦兹和爱因斯坦使用了“复杂的计算”来证明麦克斯韦方程的普通分量形式在这些变换下保持不变。索末菲的目标是在闵可夫斯基的工作基础上,展示如何以不变的四维形式写出这些方程。

In his two 1910 papers on relativity, Sommerfeld introduced the term “four-vector” for 4-D vectors in space-time. He emphasised the significance of the invariance of the space-time interval, for it expresses the constancy of the speed of light (as you can see in the endnote24). Poincaré, Lorentz, and Einstein knew this, too, of course. But Sommerfeld also pointed out the importance of invariant symbolism in adapting Maxwell’s equations to space-time, noting “the complicated calculations” that Lorentz and Einstein had used in order to show that the ordinary component form of Maxwell’s equations stayed the same under these transformations. Instead, Sommerfeld aimed to build on Minkowski’s work, showing how to write these equations in invariant four-dimensional form.

他没有找到第 8 章中所示的三维“Heaviside”矢量形式的方程的四维类似物;相反,他表明,我们现在所说的“张量”是表达四维麦克斯韦方程的形式不变性所必需的。

He didn’t find a 4-D analogue of the 3-D “Heaviside” vector form of the equations shown in chapter 8; rather, he showed that what we now call “tensors” are needed to express the form-invariance of the 4-D Maxwell equations.

您可以使用张量表达更多内容!

YOU CAN SAY EVEN MORE WITH TENSORS!

张量当时还是一个新概念,我们很快就会探索其有趣的起源——但为了了解它的精髓,索末菲是这样解释它们的。使用矢量,你可以探索线(或箭头)的几何形状;例如,在三维空间中,垂直线的特征为矢量方程ab = 0,平行线的特征为a × b = 0(如你从图 9.1标题中给出的点积和叉积的几何定义中看到的那样)。但有时你还需要描述平面,它们有一个额外的特征:空间方向。这由平面“法线”的方向定义,该方向用垂直于(或法线于)平面的单位矢量表示。但为了更清楚地了解平面与张量的关系,我们可以回到一个令人惊讶的来源:麦克斯韦 1873 年的《分子与分子动力学论文》

Tensors were a new concept then, whose intriguing origin we’ll explore soon—but to get the flavor, this is how Sommerfeld explained them. With vectors you are exploring the geometry of lines (or arrows); for example, in three dimensions, perpendicular lines are characterised by the vector equation ab = 0, and parallel lines by a × b = 0 (as you can see from the geometric definitions of the dot and cross products given in the caption to fig. 9.1). But sometimes you also need to describe planes, and they have an additional feature: orientation in space. This is defined by the direction of the plane’s “normal,” which is represented by a unit vector perpendicular (or normal) to the plane. But to get a clearer idea of what planes have to do with tensors, we can go back to a surprising source: Maxwell’s 1873 Treatise.

想象一个浸没在水中的盒子——也许是理想化的船体。由于作用于其各个面上的力平衡,它在水压下仍能保持形状。从桥梁、飞机到微小晶体,各种刚体都能平衡这些“应力”。(“应力”是单位面积上的力。)当然,在可变形或弹性材料中,力也可能不平衡,但工程师还需要考虑潜在的额外应力影响,例如桥梁上的交通繁忙或大海冲击船舶。然后,有些情况下你想移动或旋转物体,因此你又需要力的净不平衡。麦克斯韦的朋友汤姆森是数学研究这些情况的先驱之一,他在 1856 年的一篇关于弹性的论文中对此进行了分析。更早的时候,奥古斯丁·柯西在 1828 年发表了关于平衡物体应力的奠基性论文。25然而,麦克斯韦想要分析浸入电磁场的磁化物体所受的力。他通过类比浸入的盒子,考虑了作用在磁化物体小立方体每个面上的力。(换句话说,他考虑了一个可以在整个物体上积分的“体积元素”。)汤姆森和柯西也使用了类似的方法。但麦克斯韦做了一些特别的事情,他不仅将这些分析扩展到电磁学,而且还用一个有两个指标的符号来表示应力。

Imagine a box submerged in water—an idealised version of the hull of a ship, perhaps. It keeps its shape under the pressure of the water because of the balance of forces acting on each of its faces. All sorts of rigid bodies, from bridges and airplanes to tiny crystals, balance these “stress” forces. (“Stress” is the force per unit area.) There are also many situations where the forces may not balance—in deformable or elastic materials, of course, but engineers also need to account for potential additional stress impacts, such as heavy traffic on a bridge or heavy seas battering a ship. Then there are situations where you want to move or rotate the body, so again you want a net imbalance of forces. Maxwell’s friend Thomson was one of the pioneers in the mathematical study of these situations, which he analysed back in 1856 in a paper on elasticity. Even earlier, Augustin Cauchy had published his foundational 1828 paper on stress in bodies in equilibrium.25 Maxwell, however, wanted to analyse the forces on a magnetised object immersed in an electromagnetic field. By analogy with the immersed box, he considered the forces acting on each face of a little cube of the magnetised body. (In other words, he considered a “volume element” that can be integrated through the whole body.) Thomson and Cauchy had used a similar approach. But Maxwell did something special, not only by extending these analyses to electromagnetism but also by representing stress with a symbol with two indices.

在数学中,“索引”在此上下文中指的是标签,而不是幂。例如,向量的分量通常用一个索引来表示,如a = ( a 1 , a 2 , a 3 );索引指的是向量的三个轴,测量分量。麦克斯韦认识到,应力是“另一种与空间方向相关但不是矢量的物理量”。他说,这是因为在三维空间中,矢量有三个分量,但应力需要九个分量(如图9.4所示)。因此,他将应力分量写为P hk,解释说第一个标签h表示应力作用的表面(即法线与h轴平行的表面),第二个标签显示产生应力的力的方向。h 有三种选择——每个空间维度一个,即每个坐标轴一个——k 有三种选择有九种可能的组合,九个分量各一个。26

In maths an “index” refers in this context to a label, not a power. For example, the components of vectors are generally written with one index, as in a = (a1, a2, a3); the indices refer to the three axes from which the components are measured. Maxwell recognised that stress was an example of “physical quantities of another kind which are related to directions in space, but which are not vectors.” He said that’s because in 3-D space a vector has three components, but a stress needs nine of them (as you can see in fig. 9.4). So, he wrote the components of stress as Phk, explaining that the first label, h, indicates the surface on which the stress is acting—it is the one whose normal is parallel to the h-axis—and the second label shows the direction of the force producing the stress. There are three choices for h— one for each dimension in space, that is, one for each coordinate axis—and three choices for k. That’s nine possible combinations, one for each of the nine components.26

图像

图 9.4。作用于立方体表面的基本应力分量。麦克斯韦指出,如果P hk = P kh,应力就不会产生旋转(他感兴趣的是磁力产生的旋转——正如旋度方程所表达的那样)。事实上,该图表明,如果P yx > P xy,则右侧表面将被拉向前方,立方体将开始旋转。

FIGURE 9.4. The fundamental stress components acting on the faces of a cube. Maxwell noted that if Phk = Pkh, the stresses won’t produce a rotation (he was interested in the rotation produced by magnetism—as expressed in his curl equation). Indeed, the diagram suggests that if, say, Pyx > Pxy, the right-hand face will be pulled forward, and the cube will begin to rotate.

麦克斯韦在这里所做的是,对一个他简称为“应力”的单一量的分量给出一个简明的定义,但现在这个量被称为“应力张量”——就像分量 ( a 1 , a 2 , a 3 ) 形成一个单一的矢量a一样。令人着迷的是,张量当时还没有被发明——至少,还没有被发明为与矢量同等的数学对象。然而,尽管柯西和汤姆森没有麦克斯韦的简洁而通用的符号,但他们也接近这个想法。

What Maxwell had done here was to give a concise definition of the components of a single quantity he referred to simply as “stress,” but which is now called the “stress tensor”—just as the components (a1, a2, a3) form a single vector, a. The fascinating thing about this is that tensors hadn’t yet been invented—at least, not as mathematical objects on a par with vectors. Yet Cauchy and Thomson had come close to the idea, too, although they didn’t have Maxwell’s concise and general notation.

图像

图 9.5。需要两个向量来生成应力张量。首先,向量n表示应力作用于表面的方向——我将其标记为x,以表明此处法线的相关分量平行于x轴。其次,向量表示力的强度和方向:标签y表示作用于此表面的力的相关分量与y轴平行。

FIGURE 9.5. You need two vectors to make a stress tensor. First, the vector n giving the direction of the surface on which the stress is acting—I’ve labeled it with an x to indicate that here the relevant component of the normal is parallel to the x-axis. Second, the vector giving the strength and direction of the force: the label y indicates that the relevant component for the force acting on this face is in the direction parallel to the y-axis.

从现代观点来看,这里的关键思想是需要两个向量来形成应力张量:一个向量表示应力作用的表面方向,另一个向量表示力本身。你可以在图 9.5中看到这一点,这是麦克斯韦对其分量P hk的定义的另一种表达方式。当数学家进入该领域时,他们会将这个概念的本质转化为更精确、更通用的张量定义。但你已经可以看到,有九个分量的张量(例如应力)可以存储比三维矢量多得多的信息。

From a modern point of view, the key idea here is that you need two vectors to form a stress tensor: one to represent the direction of the surface on which the stress is acting, and one to represent the force itself. You can see this in figure 9.5, which is another way of expressing Maxwell’s definition of his components Phk. When the mathematicians enter the field, they’ll turn the essence of this concept into a much more precise and more general definition of a tensor. But already you can see that with nine components, a tensor such as stress can store much more information than a 3-D vector can.

到 1910 年索末菲写作时,张量还没有进入主流,但他明白张量远不止是用来表示应力的一种方法。它们不一定与平面有关,因为它们可以适应任意数量的维度和应用。索末菲顺便提到了格拉斯曼,但他的重点是他的朋友闵可夫斯基,以及用时空语言重写麦克斯韦方程。为此,他效仿闵可夫斯基将EB的分量重写为二指标量——基本上给出了麦克斯韦方程的现代张量形式。27

By 1910 when Sommerfeld was writing, tensors hadn’t yet found their way into the mainstream, but he understood that they are far more versatile than just ways of representing stresses. And they don’t necessarily have anything to do with planes, for they can be adapted to any number of dimensions and applications. Sommerfeld referred in passing to Grassmann, but his focus was on his friend Minkowski, and on rewriting Maxwell’s equations in the language of space-time. To do this, he followed Minkowski in rewriting the components of E and B as two-index quantities—giving essentially the modern tensor form of Maxwell’s equations.27

我们稍后会看到这些美丽的方程。不过,首先我们需要进一步深入研究张量的演变。特别是,我们将了解它们如何编码和扩展不变性的概念——以及爱因斯坦为什么需要它们来完成他的杰作。但我们也会稍微回顾一下过去,看看非欧几里得几何与我们的故事有什么关系。因为通往张量的道路很漫长——而且,正如我们在向量中看到的那样,它是从许多令人惊讶的方向通过创造性的见解构建而成的。

We’ll see these beautiful equations later. First, though, we need to delve further into the evolution of tensors. In particular, we’ll find out how they encode and extend the idea of invariance—and why Einstein needed them for his masterpiece. But we’ll also go back in time a little, to find out what non-Euclidean geometry has to do with our story. For the road to tensors is a long one—and, as we’ve seen with vectors, too, it was built with creative insights from many surprising directions.

(10)弯曲空间和不变距离

(10) CURVING SPACES AND INVARIANT DISTANCES

走向张量

On the Way to Tensors

当爱因斯坦开始思考如何将狭义相对论推广到广义相对论(其中相对运动不必是恒定的)时,他需要一套全新的数学工具包。而最合适的人选莫过于他来自瑞士理工学院的老朋友马塞尔·格罗斯曼。格罗斯曼从未逃过闵可夫斯基的数学课!他还是一位一流的数学家和忠实的朋友。事实上,正是通过格罗斯曼的家庭,贫困的爱因斯坦才在专利局获得了那份救命的工作。格罗斯曼本人一毕业就被理工学院聘为博士生和导师助理,七年后,他成为那里的数学教授。

When Einstein began wrestling with the problem of how to extend the special theory of relativity to a general one, where the relative motion didn’t have to be constant, he needed a whole new mathematical toolkit. And who better to call on than his old friend from the Swiss “Poly,” Marcel Grossmann. Grossmann had never cut Minkowski’s maths classes! He was also a first-rate mathematician and a loyal friend. In fact, it was through Grossmann’s family that the impoverished Einstein had landed that life-saving job at the patent office. Grossmann himself had been given a place at the Poly— as a PhD student and an assistant to his supervisor—as soon as he graduated, and seven years later he’d become professor of mathematics there.

爱因斯坦多年来一直想找人收他为博士生,但他得到的都是拒绝——部分原因是他在主流学术界“老庸人”中的傲慢名声。尽管如此,他还是坚持了下来,1905 年夏天,他终于凭借一篇关于测量分子的里程碑式论文获得了苏黎世大学的博士学位。他把这篇论文献给了格罗斯曼,以感谢他在学生时代的慷慨——借给他他逃课的笔记帮助他找到了工作。与他的教授不同,格罗斯曼从一开始就看到爱因斯坦注定会成为伟人。1

Einstein had tried for years to find someone to take him on as a doctoral student, but all he’d got for his trouble were rejections—partly because of his sassy reputation among the “old philistines” of mainstream academia. Still, he persevered, and in the summer of 1905, he’d finally had some luck with a landmark paper on measuring molecules: it earned him a doctorate from the University of Zurich. He dedicated it to Grossmann, in appreciation of his generosity in their student days—from lending him notes from lectures he’d skipped to helping him find a job. Unlike his professors, Grossmann had seen from the outset that Einstein was destined for greatness.1

图像

马塞尔·格罗斯曼 (Marcel Grossmann),1909 年。苏黎世联邦理工学院图书馆,Bildarchiv/摄影师未知/Portr_01239。公共领域。

Marcel Grossmann, 1909. ETH-Bibliothek Zürich, Bildarchiv/Photographer unknown/Portr_01239. Public domain.

1911 年,格罗斯曼开始想办法让这位如今已声名远扬的朋友重返母校——当时这所学校刚刚更名为瑞士联邦理工学院(仍以德语缩写 ETH 命名)。玛丽·居里和亨利·庞加莱等名人都曾为爱因斯坦写过热情洋溢、见解深刻的推荐信。1912 年,爱因斯坦被任命为 ETH 新设立的理论物理学教授。或许,他很享受这种光荣的回归。他当然很享受再次与老同学共事的机会,就像他们以前在学生时代一起学习和探索想法一样,在附近的 Café Metropole 抽烟斗、喝咖啡。

In 1911, Grossmann began seeking ways to bring his now-famous friend back to their old school—which had recently been renamed the Swiss Federal Institute of Technology (still known as ETH for its German acronym). Marie Curie and Henri Poincaré were among those who wrote glowing and perceptive testimonials for Einstein, and in 1912 he was appointed to the newly established chair of theoretical physics at the ETH. Perhaps he savored such a glorious return. He certainly relished the chance to work once again with his old classmate, the way they used to study and explore ideas together as students, smoking their pipes and drinking coffee at the nearby Café Metropole.

在接下来的两年里,这两位朋友密切合作,完成了为一般理论建立数学基础的艰巨任务相对论。爱因斯坦已经意识到,在推广狭义相对论时,他所研究的无非是一套全新的引力理论。毕竟,相对运动不恒定的一个例子是,当一个观察者因重力而下落时——比如在自由落体的电梯中——而另一个观察者固定在地面上:下落的观察者相对于地面观察者正在加速,因此他们的相对速度不是恒定的。然而,1912 年,当爱因斯坦和格罗斯曼开始合作时,在爱因斯坦找到他的广义相对论之前,还有两个主要的数学问题需要解决。

For the next two years, the two friends collaborated closely on the herculean task of building mathematical foundations for the general theory of relativity. Einstein had already realised that in generalising the special theory, he was working on nothing less than a whole new theory of gravity. After all, an example where the relative motion isn’t constant is when one observer is falling because of gravity—in a free-falling elevator, say—and the other is fixed to the ground: the falling observer is accelerating relative to the ground-based one, so their relative speed isn’t constant. In 1912, however, when Einstein and Grossmann began their collaboration, there were still two major mathematical problems to solve before Einstein could find his general theory.

首先,如何将物理定律从狭义理论转移到广义理论;其次,如何在弯曲的时空中找到适当的“距离”或区间测度(我们将在第 12 章中看到它为什么是弯曲的),类似于我们在上一章中看到的平坦的闵可夫斯基度量。正如爱因斯坦后来回忆的那样,“我们发现,解决问题 1 的数学方法就在里奇和列维-奇维塔的绝对微分学中,也就是在张量微积分中。”“至于问题 2,”他继续说,答案就在伯恩哈德·黎曼对曲面的研究之中。2

First, how to transfer the laws of physics from the special theory to the general one; and second, how to find the appropriate “distance” or interval measure in curved space-time (we’ll see why it’s curved in chap. 12), analogous to the flat Minkowski metric we saw in the previous chapter. As Einstein later recalled, “We found that the mathematical methods for solving problem 1 lay ready in our hands in the absolute differential calculus of Ricci and Levi-Civita”—that is, in tensor calculus. “As for problem 2,” he continued, the answer lay in Bernhard Riemann’s work on curved surfaces.2

这些工具可能是“现成的”,但爱因斯坦首先必须掌握它们。正如他告诉阿诺德·索末菲的那样,“我一生中从未如此努力过。”他接着说,在格罗斯曼的帮助下,他“对数学产生了极大的敬意,而我天真地认为数学的微妙部分至今仍是纯粹的奢侈品。”闵可夫斯基一定会很激动。3

“Ready-made” these tools may have been, but first Einstein had to master them. As he told Arnold Sommerfeld, “In all my life I have never before labored [so] hard.” He went on to say that with Grossmann’s help, he had “become imbued with a great respect for mathematics, the subtle parts of which, in my innocence, I had till now regarded as pure luxury.” Minkowski would have been thrilled.3

正如我们所见,索末菲已经尝试使用张量来在平坦的四维时空中写出麦克斯韦方程——而麦克斯韦本人也曾使用二指标张量来描述普通三维空间中的应力。格雷戈里奥·里奇和图利奥·列维-奇维塔所做的是开发一种可以处理弯曲空间的微积分——而格罗斯曼和爱因斯坦是第一个将张量微积分应用于弯曲时空的人。但他们都是以黎曼的工作为基础的——而他则以他的传奇老师高斯的思想为基础。

As we’ve seen, Sommerfeld had already dabbled with tensors as a way of writing Maxwell’s equations in flat 4-D space-time—and Maxwell himself had used two-index tensorial quantities to describe stresses in ordinary 3-D space. What Gregorio Ricci and Tullio Levi-Civita did was to develop a calculus that could handle curved spaces, too—and it was Grossmann and Einstein who first applied tensor calculus to curved space-time. But they all built on Riemann’s work—and he’d built on the ideas of his legendary teacher, Gauss.

曲面数学:卡尔·弗里德里希·高斯

THE MATHS OF CURVED SURFACES: CARL FRIEDRICH GAUSS

格罗斯曼的博士学位论文是非欧几里得几何,该几何由高斯、亚诺什·鲍约和尼古拉·罗巴切夫斯基在 19 世纪 20 年代开创。在曲面几何中,欧几里得关于平行线永不相交的公理不再成立,这一点从地球仪上的经线最容易看出:它们在赤道处平行,但在两极相交。那么,你能明确地说出曲面上的“平行”线吗?这对于矢量分析来说是一个至关重要的问题,因为矢量加法的整个概念都基于平行四边形规则,通过保持矢量平行,将它们滑动到正确的位置(如图3.1所示)。

Grossmann had done his PhD on non-Euclidean geometry, which had been pioneered by Gauss, Janos Bolyai, and Nicolai Lobachevsky in the 1820s. With the geometry of curved surfaces, Euclid’s axiom about parallel lines never meeting no longer holds, and you can see this most readily with lines of longitude on the globe: they’re parallel at the equator but they meet at the poles. So, can you ever speak definitively about “parallel” lines on a curved surface? This is a crucial question for vector analysis because the whole concept of addition of vectors is based on the parallelogram rule, where you slide vectors to the correct position by keeping them parallel (as in fig. 3.1).

这个问题的答案来得晚得多,我们见到列维-奇维塔时就会知道。但他受益于里奇,而里奇又受益于高斯在相关问题上的工作。在日常生活中,我们用直尺沿着两点之间的直线测量两点之间的距离,毕达哥拉斯定理给出了距离测量或“度量”。但如果曲面上没有直线,距离公式是什么?高斯决定,解决方案是查看非常接近的点。

The answer to this question came much later, as we’ll see when we meet Levi-Civita. But he was indebted to Ricci, who, in turn, was indebted to Gauss’s work on a related problem. In everyday life, we measure the distance between two points with a straight ruler laid along the straight line between the points, and Pythagoras’s theorem gives the distance measure or “metric.” But if there are no straight lines on a curved surface, what is the distance formula? The solution, Gauss decided, was to look at points that are very close together.

高斯对测量距离了如指掌。从十几岁起,他就对大地测量学(对地球形状和表面面积的数学分析)和三角测量法(测量方法)产生了兴趣。从那时起,他参与了各种政府和军事测量项目——战争似乎永无休止。事实上,正是由于 1803 年至 1815 年间肆虐的拿破仑战争,索菲·热尔曼才最终向高斯透露了自己的真实身份。正如我们在第 3 章中看到的那样,她曾以笔名勒布朗先生给高斯写信以寻求数学建议。但当法国军队占领了离高斯家不远的一个普鲁士小镇时,她非常担心高斯,于是说服了一位家庭朋友(也是一名军事将领)检查他的安全。最让高斯惊讶的是敌军士兵的礼节性拜访——而当士兵说他是受杰曼小姐的委托来拜访的时,他更加困惑了!

Gauss knew all about measuring distances firsthand. Ever since he was a teenager he’d been interested in geodesy—the mathematical analysis of the shape of Earth and the area of its surface—and in triangulation, the method used for surveying. Since then, he’d worked on various government and military surveying projects—war was still seemingly neverending. In fact, it was because of the Napoleonic Wars, which had raged during the years 1803–15, that Sophie Germain had finally revealed her true identity to Gauss. As we saw in chapter 3, she’d sought mathematical advice by writing to him under the pseudonym Monsieur Le Blanc. But when French forces took a Prussian town not far from Gauss’s home, she was so afraid for him that she persuaded a family friend—who was also a military general—to check on his safety. Gauss was most surprised by a courteous visit from an enemy soldier—and even more perplexed when the soldier said he’d been sent on behalf of Mademoiselle Germain!

从 1821 年到 1825 年,年逾四十的高斯带领一系列勘测探险队,绘制了整个汉诺威王国的地图。这些探险不仅需要数学知识和仪器使用方面的专业知识,还需要穿越艰难的地形和艰苦的生活。然后,在晚上,高斯必须手动处理他和他的团队收集的数据——为了帮助自己,他使用了他已经发明的最小二乘法。这是一种将线拟合到数据点的方法,或者从一堆重复测量中找到最准确的估计值——它的许多现代应用包括我在机器学习中提到的线性回归模型。高斯几乎和两个世纪前那位神秘的伊丽莎白时代的托马斯·哈里奥特一样不愿发表论文,因为他们都是完美主义者——因此最小二乘法以及高斯的无数其他发现后来被其他人重新发明。4

From 1821 to 1825, Gauss, now in his forties, led a series of surveying expeditions mapping the entire kingdom of Hanover. These expeditions required not just mathematical know-how and expertise with instruments but also traveling through difficult terrain and living rough. Then, at night, Gauss would have to process by hand the data he and his team had collected—and to help himself he used the method of least squares that he’d already invented. This is a method for fitting a line to data points, or for finding the most accurate estimate from a bunch of repeated measurements— and among its many modern applications are the linear regression models I mentioned in connection with machine learning. Gauss was almost as reluctant to publish as that enigmatic Elizabethan Thomas Harriot had been two centuries earlier, for both men were perfectionists—and so the least squares method, and countless other of Gauss’s discoveries, were reinvented later by others.4

无论如何,在经历了所有这些实践经验后,高斯准备发表他 1828 年的开创性论文曲面的一般研究》。事实上,高斯创造了度量的概念,也就是我们在上一章中遇到的距离度量,在那里我将平面三维欧几里得度量写为

Anyway, after all this hands-on experience Gauss was ready to present his seminal 1828 paper, Disquisitiones Generales circa Superfices Curvas (General Investigations of Curved Surfaces). In fact, it is Gauss who created the concept of the metric, the distance measure we met in the previous chapter, where I wrote the flat 3-D Euclidean metric as

2+2+2

x2+y2+z2.

我假设x, y, z是位置矢量的分量,表示从原点到点 ( x, y, z )的直线距离,但任意两点 ( x 1 , y 1 , z 1 ) 和 ( x 2 , y 2 , z 2 ) 之间的距离由下式给出:

I assumed that x, y, z were the components of the position vector giving the distance along the straight line from the origin to the point (x, y, z), but the distance between two arbitrary points (x1, y1, z1) and (x2, y2, z2) is given by

212+212+212

x2x12+y2y12+z2z12.

高斯的想法是,如果曲面上的两个点足够接近,以至于x 2x 1非常小,其他坐标差也类似,那么它们之间的小块曲面近似平坦。你可以直观地想象一个橙子或一个球上的两个相邻点——从更大的尺度上看,这就是为什么我们可以说曲面地球上有一块“平坦”的土地。如果两个点无限接近,那么它们之间的曲面在所有意图和目的上都是平坦的,它们之间的线将是直线。用现代术语来说,曲面是“局部”的平面。这与用切线近似曲线的一小部分的想法相同。您可以使用这条直切线来近似曲线上附近点的距离,如图10.1所示——只是对于曲面您有一个切平面而不是切线。

Gauss’s idea was that if two points on a curved surface are close enough together, so that x2x1 is very small, and similarly for the other coordinate differences, the little piece of surface between them is approximately flat. You can see this intuitively by imagining two nearby dots on an orange or a ball—and on a larger scale it’s why we can speak of a patch of “flat” land on the curved Earth. If your two points are infinitesimally close together, the surface between them will be flat to all intents and purposes, and the line between them will be straight. In modern terms, the surface is “locally” flat. It’s the same idea as when a small section of a curved line is approximated by a tangent line. You can use this straight tangent line to approximate the distance to a nearby point on the curve, as in figure 10.1—except that with the curved surface, you have a tangent plane rather than a tangent line.

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图 10.1。当曲线或曲面上的两点非常接近时,它们之间的距离AB大约等于直线距离AB′。例如,在计算机图形学中,曲线可以由微小的切线段构建而成。

FIGURE 10.1. When two points on a curved line, or a curved surface, are very close together, the distance between them, AB, is approximately the straight-line distance AB′. In computer graphics, for example, curved lines can be built from tiny tangential segments.

这意味着你可以在曲面上使用欧几里得距离测量,只不过不是x 2x 1 , y 2y 1 , z 2z 1,而是需要微积分中使用的无穷小距离:dx, dy, dz。换句话说,不是

What this means is that you can use the Euclidean distance measure on a curved surface, except that instead of x2x1, y2y1, z2z1, you need the kind of infinitesimal distances used in differential calculus: dx, dy, dz. In other words, instead of

2+2+2或者212+212+212

x2+y2+z2orx2x12+y2y12+z2z12,

高斯表明,一般的距离测量或度量是

Gauss showed that the general distance measure or metric is

ds=d2+d2+d2

ds=dx2+dy2+dz2,

其中,曲面上线的长度用s表示。这个表达式通常是平方的,为了方便书写,省略了括号,因此曲面的欧几里得度量通常写为

where the length of a line on the surface is denoted by s. Often this expression is squared, and, taking liberties with notation, the brackets are left out to make it easier to write—so the Euclidean metric for the surface is generally written as

ds2 = dx2 + dy2 + dz2

ds2 = dx2 + dy2 + dz2.

(在高斯之后 80 年,闵可夫斯基知道用这种微分符号来表示他的四维时空度量:5

(Writing eighty years after Gauss, Minkowski knew to use this kind of differential notation for his 4-D space-time metric:5

ds 2 = dx 2 + dy 2 + dz 2c 2 dt 2。

ds2 = dx2 + dy2 + dz2c2 dt2.)

如果您不熟悉这些,您可能会认为将距离度量作为细直线的长度来讨论很好,但是当您想要测量更长的曲线距离时,您该怎么做?从表面上看,您必须沿着表面的曲线首尾相连地放置无穷小标尺。幸运的是,莱布尼茨微分符号可以非常清楚地解释如何更简单地回答这个问题:只需对ds进行积分即可找到距离s。我在图 2.3b中使用了此度量的 2-D 版本执行了此操作。如果您熟悉这些,我希望您会同意,回顾并看看我们今天所学的数学来自何处(在本例中是强大的高斯)是很有趣的。但他并没有使用“度量”一词;相反,他将这种距离度量称为表面的“线性元素”——它仍然经常被称为“线元素”。

If all this isn’t familiar to you, you might be thinking that it’s all very well to talk about distance measures as the length of tiny straight lines, but what do you do when you want to measure longer curved distances? On the face of it, you’d have to lay down your infinitesimal rulers end-to-end along the curved line across the surface. Fortunately, the Leibnizian differential notation makes it beautifully clear how to answer this question much more simply: just integrate ds to find the distance s. I did this in figure 2.3b, using the 2-D version of this metric. If you are familiar with this, I hope you agree that it’s interesting to look back and see where the maths we learn today comes from—in this case, the mighty Gauss. He didn’t use the term “metric,” though; instead, he called this distance measure the “linear element” of the surface—and it’s still often called the “line element.”

比如说,球的曲面只有二维,所以你也许会想,为什么三维平面空间和二维曲面上的一小块区域之间的度量看起来是一样的。当我们观察球的表面时,我们是从外部观察的,所以我们将表面上的点视为三维空间中的点。但正如你在图 10.2中看到的那样,与图 10.1类似,在表面上测量的两个点之间的无穷小距离实际上与在空间中测量的直线距离相同。

The curved surface of a ball, say, is only two-dimensional, so you might also be wondering why the metric looks the same for flat 3-D space and a small patch on a 2-D curved surface. When we look at the ball’s surface, we are looking at it from the outside, so we see the points on the surface as points in 3-D space. But as you can see in figure 10.2, and analogously in figure 10.1, the infinitesimal distance between two of those points, measured on the surface, is virtually the same as the straight-line distance measured in space.

因此,为了强调表面固有的二维性质(超级智能蚂蚁或二维外星人会这样看待它),高斯效仿莱昂哈德·欧拉,用两个“曲线”坐标(他称之为pq )来参数化表面。例如,在地球表面,p轴可能是赤道周围的纬线,q轴可能是格林威治经度子午线。爬过这个表面的智能蚂蚁外星人会从这两个轴进行测量——它不会意识到存在第三维。

So, to highlight the intrinsic, 2-D nature of the surface—the way that a super-intelligent ant or 2-D alien would see it—Gauss followed Leonhard Euler’s lead by parameterising surfaces in terms of two “curvilinear” coordinates, which he called p and q. On the surface of Earth, for example, the p-axis could be the line of latitude around the equator, and the q-axis the meridian of longitude through Greenwich. An intelligent ant-alien crawling over this surface would take measurements from these two axes—it wouldn’t be aware there was a third dimension.

高斯证明,当你这样做时,二维表面上的度量将变成

Gauss showed that when you do this, the metric on the 2-D surface becomes

图像

图 10.2 . 相距无穷小距离的两点之间的曲线距离几乎与它们之间的直线距离相同,在三维空间中测量时就好像表面不存在一样。

FIGURE 10.2. The curved distance between two points that are separated by an infinitesimal distance is almost the same as the straight-line distance between them, measured in 3-D space as if the surface weren’t there.

dx 2 + dy 2 + dz 2 = Edp 2 + 2 Fdpdq + Gdq 2 ,

dx2 + dy2 + dz2 = Edp2 + 2Fdpdq + Gdq2,

其中E、F、G是pq的函数。(你可以在下一个尾注中看到他是如何做到的。)他还以精湛的技艺证明了,这个表达式中的系数E、F、G及其导数包含了我们的二维生物需要知道的一切,以弄清表面的内在几何形状。

where the E, F, G are functions of p and q. (You can see how he did it, in the next endnote.) In a virtuoso feat he also showed that the coefficients E, F, G in this expression, along with their derivatives, contain all that our 2-D creature would need to know to figure out the intrinsic geometry of the surface.

例如,外星人蚂蚁能够分辨出自己是在平坦空间中爬行,其中三角形内角和为 180°,是在球体等正弯曲空间中爬行,其中内角和大于 180°,还是在马鞍形表面等负弯曲表面上爬行,其中内角和小于 180°(如图10.3所示)。

For example, the ant-alien would be able to tell if it were crawling over a flat space, where the angles in a triangle add up to 180°, a positively curved space such as a sphere, where they add up to more than 180°, or a negatively curved surface such as that of a saddle, where they add to less than 180° (as in fig. 10.3).

这是因为高斯用弯曲三角形的面积和 180° 与三角形内角和之来定义“固有曲率” 。他当时并不知道这一点,但对于球体,托马斯·哈里奥特在绘制地图和探索地球表面的过程中已经发现了这个公式:

That’s because Gauss defined the “intrinsic curvature” in terms of the area of the curved triangle and the difference between 180° and the triangle’s angle sum. He didn’t know it, but for the case of a sphere Thomas Harriot had already discovered this formula, during his own work on mapmaking and navigating the surface of the earth:

α+β+γ-π三角形面积=1r2

α+β+γ-πArea of triangle=1r2,

其中三角形的角度用弧度 α、β、γ 表示,π 弧度等于 180°,r是球体的半径,并且1r2是其“高斯”或内在曲率。由于哈里奥特没有发表他的成果,几十年后,阿尔伯特·吉拉德重新发现并发表了这一成果——但高斯的版本更为复杂,并不仅限于球体。此外,高斯还表明,该公式中的面积和角度都可以仅通过度量来求得。这乍一看相当令人惊讶。事实上,高斯非常兴奋,他将这一发现称为他的“非凡定理”

where the triangle’s angles are designated by α, β, γ radians, π radians equal 180°, r is the radius of the sphere, and 1r2 is its “Gaussian” or intrinsic curvature. Because Harriot didn’t publish his result, Albert Girard rediscovered and published it several decades later—but Gauss’s version was more sophisticated, and not just restricted to spheres. What’s more, Gauss showed that both the area and the angles in this formula could be found from the metric alone. This is quite astonishing at first sight. In fact, Gauss was so excited he called this discovery his theorema egregium, his “remarkable theorem.”

那么,让我再进一步解释一下这个非凡的结果。我们已经看到,度量可以告诉你一条线的长度——比如我们在第 9 章中看到的矢量的大小,或者图 2.3圆的周长。很久以前,阿基米德就计算出了球体的表面积 (4πr2 ),甚至还展示了如何求出球体圆柱段的面积。但高斯表明,实际上,任何曲面的面积都可以通过度量系数E、F、G通过表达式的面积积分来找到F2在没有微积分和度量概念的情况下,当哈里奥特推导出球面上三角形的曲面面积时——他是已知的第一个做到这一点的人,而且他需要它来证明上面的内在曲率公式——他不得不使用许多页巧妙的几何论证。

So let me unpack this remarkable result a little more. We’ve already seen that the metric tells you the length of a line—such as the magnitude of a vector, as we saw in chapter 9, or the circumference of a circle as in figure 2.3. And long ago Archimedes had worked out the surface area of a sphere (4πr2), and had even shown how to find the area of cylindrical segments of the sphere. But Gauss showed that the area of any curved surface could, in fact, be found from the metric coefficients, E, F, G, via the surface integral of the expression EGF2. Without calculus and the concept of a metric, when Harriot derived the curved area of a triangle on a sphere—he was the first known person to do this, and he needed it to prove the intrinsic curvature formula above—he had to use many pages of ingenious geometrical arguments.

请注意,高斯也花了很多篇幅来建立曲率和度量之间的基本关系,但哈里奥特只解决了球面三角形问题,而高斯则为任何表面上任何形状的面积指明了方向。至于高斯和哈里奥特的曲率公式中所需的角度,令人惊讶的是,E、F、G是与所示坐标线pq相切的单位向量的标量积图 10.4中— 标量积的几何公式​​给出了这些向量之间夹角的余弦。当然,1828 年高斯还不知道向量和标量积 — 至少在形式上不知道 — 但他有等价的坐标形式。我在尾注中解释了他是如何做到的。6

Mind you, it took Gauss many pages, too, to establish these fundamental relationships between curvature and the metric, but whereas Harriot had solved only the spherical triangle problem, Gauss showed the way for areas of any shape on any surface. As for the angles needed in Gauss’s and Harriot’s curvature formula, amazingly it turns out that E, F, G are scalar products of the unit vectors tangent to the coordinate lines p and q shown in figure 10.4—and the geometric formula for scalar products gives the cosine of the angle between these vectors. Of course, in 1828 Gauss didn’t know about vectors and scalar products—at least not in the formal sense— but he had the equivalent coordinate form, nonetheless. I’ve explained how he did it in the endnote.6

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图 10.3 . 平整纸张上三角形的角 α、β、γ 之和为 180° 或 π 弧度,但在球体上,它们之和大于 π,而在马鞍形上,它们之和小于 π。

FIGURE 10.3. The angles α, β, γ in a triangle on a flat piece of paper add up to 180° or π radians, but on a sphere, they add to more than π, and on a saddle they add to less.

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图 10.4由三条大圆坐标线围成的球面三角形。两条大圆坐标线之间的夹角就是它们的切向量之间的夹角,如箭头所示。在球面上,所有三个角都是直角。

FIGURE 10.4 A spherical triangle bounded by three great circle coordinate lines. The angle between two such lines is the angle between their tangent vectors, as illustrated by the arrows. On a sphere all three angles are right angles.

值得注意的是,高斯和哈里奥特在曲率方面取得突破的动力都是实用的——绘制地图和导航。所以,这一切并不只是蚂蚁和外星人的专利。我们也可以使用度规来确定我们居住的表面的曲率——我们不必前往太空就可以从外部观察地球。这真的很了不起,就好像一个小小的方程式ds 2 = Edp 2 + 2 Fdpdq + Gdq 2,就编码了整个表面的全视上帝视角。当我们遇到格雷戈里奥·里奇时,我们会看到,度规中微分的系数——比如这里的E、F、G——原来是张量的分量,张量是矢量的下一步。在爱因斯坦的手中,它将成为整个宇宙的钥匙。

It’s telling that the impetus for both Gauss and Harriot to make their breakthroughs on curvature was practical—mapmaking and navigation. So, all this isn’t just for ants and aliens. We, too, can use the metric to determine the curvature of the surface we live on—we don’t have to travel into space to see Earth from the outside. It really is remarkable, as if one tiny equation, ds2 = Edp2 + 2Fdpdq + Gdq2, encodes an all-seeing god’s-eye view of the whole surface. And as we’ll see when we meet Gregorio Ricci, the coefficients of the differentials in a metric—such as the E, F, G here— turn out to be the components of a tensor, the next step on from a vector. In Einstein’s hands, it will be a key to the whole cosmos.

从地图绘制到黑洞:不变性、拓扑结构和“直”线

FROM MAPMAKING TO BLACK HOLES: INVARIANCE, TOPOLOGY, AND “STRAIGHT” LINES

爱因斯坦还将借鉴高斯强调的另一个概念:不变性的概念。事实上,不变性对于经济和权力至关重要张量方程的不变性,就像我们在上一章中看到的向量一样。特别是,拥有一个不变的距离度量非常重要,因为欧几里得距离或时空中事件之间的闵可夫斯基间隔如果从不同的角度测量(例如从旋转的框架或洛伦兹变换的框架)将保持不变。我们之前看到了这样的例子,它们说明了我们上面遇到的欧几里得度量和闵可夫斯基度量分别在旋转和洛伦兹变换下不变的事实。

Einstein will also draw on something else that Gauss highlighted: the concept of invariance. In fact, invariance is critical to the economy and power of tensor equations, as we saw for vectors in the previous chapter. In particular, it’s important to have an invariant distance measure, in the sense that the Euclidean distance, say, or the Minkowski interval between events in space-time, would remain the same if it were being measured from a different point of view—from a rotated frame, for example, or a Lorentztransformed one. We saw examples of this earlier, and they illustrate the fact that the Euclidean and Minkowski metrics we met above are invariant under rotations and Lorentz transformations, respectively.

1828 年,洛伦兹变换的出现已经是半个多世纪以后的事情了,但高斯确实证明了他的二维曲线度量——他称之为“线性元素”或“曲率度量”——在将表面弯曲成不同形状时是不变的(或者用他的话来说,“不变”)。至少,如果你没有撕开或切割表面,情况确实如此:例如,将一张纸卷成圆柱体,或将足球塑造成橄榄球。后一种塑造就是熔融和流体地球的情况,由于自转,地球在赤道处隆起,但其总体积没有改变。

In 1828 Lorentz transformations were more than half a century into the future, but Gauss did show that his 2-D curvilinear metric—or what he called the “linear element” or “measure of curvature”—was invariant (or as he put it, “unchanged”) when you bent the surface into a different shape. At least, this was true if you didn’t tear or cut the surface: rolling up a piece of paper to form a cylinder, for example, or moulding a soccer ball into a football. This latter kind of moulding is what happens with the molten and fluid Earth, which bulges at the equator because of its rotation yet doesn’t change its total volume.

在这种弯曲或挤压下,度量不变性与圆柱体的曲面本质上是“平坦”的反直觉思想有关。(只有从外部看时它才是弯曲的,因此这种曲率是“外在的”。)为了看到这种不变性,想象一张平整的纸上画了一条直线;就我们的二维蚂蚁所知,它就是那条线——长度相同,因此度量也相同——就像纸卷成圆柱体时一样。但是球体本质上是弯曲的,正如你在榨完半个橙子后可能注意到的那样:你不能在不撕裂它的情况下将半球壳压平。它只是局部平坦的,如图 10.110.2所示。所以,我们在这里讨论的不仅是坐标变换,还有拓扑——它涉及可以塑造而不会撕裂的表面和形状的属性——尽管高斯没有使用这个术语。我们在第 5 章中简要介绍了拓扑的概念,其中介绍了高斯的学生莫比乌斯和李丁在 19 世纪 40 年代的工作,当时距离高斯的论文发表已经过去了 20 年。尽管如此,高斯所展示的度量拓扑不变性解释了为什么凸起在测量距离和角度时,地球可以被视为一个球体。

The invariance of the metric under this kind of bending or squashing relates to the counterintuitive idea that the curved surface of a cylinder is intrinsically “flat.” (It is only curved when viewed from the outside, so this kind of curvature is “extrinsic.”) To see the invariance, imagine a flat sheet of paper with a straight line drawn on it; as far as our 2-D ant can tell, it is just the same line—with the same length, and therefore the same metric—as when the paper is rolled up into a cylinder. But a sphere is intrinsically curved, as you may have noticed after juicing a half-orange: you can’t flatten out the hemispherical shell without tearing it. It’s only flat locally, as in figures 10.1 and 10.2. So, we’re talking not only coordinate transformations here but also topology—which deals with properties of surfaces and shapes that can be moulded without tearing—although Gauss didn’t use this term. We saw the idea of topology briefly in chapter 5, with the work of Gauss’s students Möbius and Listing in the 1840s, which was two decades after Gauss’s paper. Nonetheless, what Gauss had shown about the topological invariance of the metric explains why the bulging Earth can be treated as a sphere when it comes to measuring distances and angles.

曲率和拓扑之间更为显著的联系在所谓的全局高斯-博内定理中得到体现。这个定理的“局部”版本只是高斯对曲率的定义,即曲面一块区域上弯曲三角形的角度和面积——哈里奥特公式是球面三角形的特例——必要时,皮埃尔·奥西安·博内于 1848 年将高斯定理扩展到圆盘等开放曲面。当你将它应用到整个(“全局”)曲面(例如整个球体)而非简单地一块区域时,拓扑就出现了。1972 年,斯蒂芬·霍金利用全局高斯-博内定理和爱因斯坦方程证明了稳态黑洞的边界或事件视界在拓扑上等同于球体。这是人类小“蚂蚁”如何坐下来,用纸笔和曲率数学探索广阔而神秘的新领域的另一个例子。前一年,霍金用数学证明了这个边界的面积永远不会减小——这一结果直到 2021 年才得到实验证实。但正是罗杰·彭罗斯在 1965 年利用拓扑学证明,如果广义相对论是正确的,那么黑洞真的应该存在于自然界中——它们不仅仅是数学产物。然后,在 2019 年,得益于事件视界望远镜背后的国际合作,我们都看到了黑洞的第一张非凡的直接图像——或者更确切地说,它大致(拓扑上!)呈圆形的阴影。7

An even more remarkable connection between curvature and topology is expressed in what’s called the global Gauss-Bonnet theorem. The “local” version of the theorem is just Gauss’s definition of curvature in terms of the angles and area in a curved triangle on a patch of the surface— with Harriot’s formula as a special case for spherical triangles—and, when needed, Pierre Ossian Bonnet’s 1848 extension of Gauss’s theorem to open surfaces such as disks. When you apply this not simply to a patch but to the whole (“global”) surface, such as a whole sphere, topology comes into it. In 1972, Stephen Hawking used the global Gauss-Bonnet theorem and Einstein’s equations to prove that the boundary, or event horizon, of a stationary black hole is topologically equivalent to a sphere. It’s another example of the way little human “ants” can sit with their pens and paper and use the maths of curvature to discover vast and mysterious new territory. The year before, Hawking had proved mathematically that the area of this boundary could never decrease—a result that wasn’t confirmed experimentally until 2021. But it was Roger Penrose who, in 1965, had used topology to prove that if general relativity is correct, then black holes really should exist in nature—they were not just mathematical artifacts. And then, in 2019, thanks to the international collaboration behind the Event Horizon Telescope, we were all treated to that extraordinary first direct image of a black hole—or rather, its roughly (topologically!) circular shadow.7

彭罗斯因证实黑洞的存在而获得了 2020 年诺贝尔物理学奖。另外两位获奖者是天文学家里恩哈德·根泽尔和安德里亚·盖兹,他们发现了银河系中心的“超大质量致密物体”——据推测是黑洞。盖兹是第四位获得诺贝尔物理学奖的女性。但让我回到高斯曲率,因为它不仅在测地学、测量学和宇宙学中有用:如今它具有广泛的实际应用,包括尖端材料科学。8

Penrose shared the 2020 physics Nobel Prize after this spectacular confirmation of the existence of black holes. The other two recipients, astronomers Rienhard Genzel and Andrea Ghez, discovered the “supermassive compact object”—assumed to be a black hole—at the centre of our galaxy. Ghez is only the fourth woman to have won a Nobel Prize for Physics. But let me come back to Gaussian curvature, for it isn’t only useful in geodesy, surveying, and cosmology: today it has a wide range of down-to-earth applications, including cutting-edge materials science.8

• • •

• • •

高斯的另一个绝妙见解为今天头条新闻背后的复杂数学铺平了道路。正如汉密尔顿意识到四元数本身就形成了一个代数系统,其规则与普通代数不同,高斯意识到弯曲的二维表面本身就是一个“空间”,就像我们生活的三维空间一样。这样的空间有两个曲线坐标轴和自己的距离度量,有自己固有的二维几何形状,我们已经看到度量是这种几何形状的关键,因为它是计算距离、角度和曲率所必需的。但它的意义远不止于此。在普通(平坦)空间中,我们有欧几里得几何形状。它是直线的几何形状,以及它们彼此之间的角度,关于这些线和角度的无数定理,两千多年来一直为我们服务。那么,高斯面临的问题是:如何在没有直线的空间中找到几何规则?他的回答非常巧妙。直线的特点是,它是普通欧几里得空间中两点之间的最短距离——因此问题变成了,曲面上两点之间的最短距离是多少?

Gauss had another brilliant insight that paved the way for the sophisticated maths behind the headlines today. Just as Hamilton realised that quaternions formed an algebraic system in their own right, with different rules from ordinary algebra, Gauss realised that a curved 2-D surface was a “space” in its own right, just like the 3-D space that we live in. Such a space, with its two curvilinear coordinate axes and its own distance measure, has its own intrinsic 2-D geometry, and we’ve already seen that the metric is the key to this geometry, for it is needed to calculate distances, angles, and curvature. But there’s even more to it than this. In ordinary (flat) space, we have Euclid’s geometry. It is the geometry of straight lines, and the angles they make with each other, and there are countless theorems about these lines and angles, which have served us well for more than two thousand years. So, the question for Gauss was this: How do you find geometrical rules in a space that has no straight lines? His answer is ingenious. The thing about a straight line is that it is the shortest distance between two points in ordinary Euclidean space—so the question becomes, what is the shortest distance between two points on a curved surface?

在球体(比如地球)的表面上,数学天文学家、地图绘制者和航海家几千年来都知道“大圆”——具有与球体相同的中心和半径的圆,例如经线和赤道。但约翰·伯努利和他的兄弟发现,这个表面上两点之间的最短距离是沿着穿过它们的大圆飞行。即使在今天,飞行员也会尽可能沿着大圆飞行,以提高飞行效率。在 18 世纪 20 年代,伯努利的前学生欧拉发明了所谓的“变分法”来寻找这条最短线的方程;它是学校微积分的复杂扩展,在微积分中,你可以将函数的导数设置为零来找到它的最大值和最小值。一个世纪后,高斯展示了如何将变分法应用于度量,以找到二维曲面上任意两点之间的最短距离。

On the surface of a sphere, like Earth, mathematical astronomers, map-makers, and navigators had known for millennia about “great circles”— circles that have the same centre and radius as the sphere, such as lines of longitude, and the equator. But it was Johann Bernoulli and his brother who’d figured out that the shortest distance between two points on this surface is along the great circle that runs through them. Even today, airplane pilots follow great circle lines where possible, to make their journeys more efficient. In the 1720s, Bernoulli’s former student Euler invented what is known as “the calculus of variations” to find the equation of this shortest line; it’s a sophisticated extension of school calculus, where you set derivatives of a function equal to zero to find its maxima and minima. A century later, Gauss showed how to apply the calculus of variations to the metric, to find the shortest distance between any two points on a 2-D curved surface.

这个“最短距离”位于一条称为“测地线”的线上——这个名字让我们想起了非欧几里得几何与古代航海的联系以及地球的大圆。爱因斯坦在重写弯曲空间的运动定律时,将惊人地利用测地线。但正如他后来回忆的那样,他和格罗斯曼首先需要了解高斯杰出的学生伯恩哈德·黎曼的工作,因为他是将高斯的分析带入更高维度的人。

This “shortest distance” lies on a line called a “geodesic”—a name that reminds us of non-Euclidean geometry’s connection with ancient navigation and Earth’s great circles. Einstein will make stunning use of geodesics when he rewrites the laws of motion for curved spaces. But as he recalled later, first he and Grossmann needed to understand the work of Gauss’s remarkable student Bernhard Riemann, for he’s the one who took Gauss’s analysis into higher dimensions.

伯纳德·黎曼接过高斯的接力棒

BERNHARD RIEMANN TAKES UP GAUSS’S BATON

19 世纪 40 年代末至 50 年代初,黎曼在高斯任教的哥廷根大学学习,但哥廷根大学并非本世纪后期那样充满活力的学术活动中心。教授们严肃而冷漠,他们的授课方式也十分老套,甚至高斯也只教授基础课程。因此,黎曼转学到柏林大学学习了几年,那里的教授也是优秀的数学家,他们讲授的课程更加前沿。尽管如此,他还是选择回到哥廷根大学跟随高斯攻读博士学位。

In the late 1840s and early 1850s, Riemann studied at the University of Göttingen where Gauss was a professor, but it wasn’t the vibrant centre of intellectual activity it would become later in the century. The professors were formal and remote, and their lectures were old-fashioned—even Gauss taught only elementary classes. So, Riemann transferred to the University of Berlin for a couple of years—the professors there were also excellent mathematicians, and they gave much more cutting-edge lectures. Still, he chose to do his PhD with Gauss back at Göttingen.

三年后,黎曼在论文中给自己设定了一个不小的挑战。这篇论文最先由威廉·金登·克利福德 (William Kingdon Clifford) 译成英文。克利福德是麦克斯韦才华横溢的年轻同事,也是乔治·艾略特的朋友,他发展了哈密顿和格拉斯曼的矢量思想,如我们在第 7 章中看到的。正如我在序言中提到的,如果你概括x、y、z,即我们熟悉的笛卡尔坐标,并写为x 1x 2x 3,那么很容易想象任意数量的维度,坐标轴为x 1x 2x 3、... 、 x n 。在这样的n维空间中,矢量分量(比如速度)可以表示为v 1v 2v 3、... 、v n 。黎曼的挑战是运用高斯在弯曲二维空间中的工作,找到弯曲n维空间的几何规则。

Three years later, Riemann set himself quite a challenge in a paper first translated into English by William Kingdon Clifford—Maxwell’s brilliant young colleague and George Eliot’s friend, who developed a synthesis of Hamilton’s and Grassmann’s vectorial ideas as we saw in chapter 7. As I mentioned in the prologue, if you generalise x, y, z, the familiar Cartesian coordinates, and write x1, x2, x3, then it’s easy to imagine as many dimensions as you like, with coordinate axes x1, x2, x3, … , xn. Vector components—of velocity, say—in such an n-D space would be denoted by, say, v1, v2, v3, … , vn. Riemann’s challenge was to adapt Gauss’s work on curved 2-D space and find the rules of geometry for curved n-dimensional space.

首先,黎曼必须定义一种在n维曲面上测量距离的方法。他将这个曲面,这个弯曲空间称为“流形”。(这是一个拓扑概念,因为黎曼追随高斯探索内在曲率。)

First, Riemann had to define a way of measuring distances on an n-dimensional curved surface. He called this surface, this curved space, a “manifold.” (It’s a topological concept, for Riemann followed Gauss in exploring intrinsic curvature.)

在介绍他是如何做到的之前,我应该提到,他并不是 19 世纪 50 年代唯一一个研究n维空间的人。例如,亚瑟·凯莱不仅是矩阵理论的发明者,也是矢量分析的激烈反对者;他还是不变理论和 n维的先驱。几何——具体来说,就是曲面投影到平坦的欧几里得空间时的几何形状,就像地面上的阴影一样。

Before we see how he did it, I should mention that he wasn’t the only one investigating n-dimensional spaces in the 1850s. Arthur Cayley, for example, wasn’t just the inventor of matrix theory, and a spirited opponent of vector analysis; he was also a pioneer in both invariant theory and n-D geometry—specifically, the geometry of a curved surface when it is projected onto flat Euclidean space, like a shadow on the ground.

说到凯莱,二十年后,也就是 1874 年,他被授予了一幅肖像,这幅肖像至今仍挂在三一学院的餐厅里,旁边是麦克斯韦的肖像,对面是牛顿的肖像(麦克斯韦于 1871 年回到剑桥,担任该校第一个科学实验室卡文迪什的第一任主任)。这次活动激发了麦克斯韦创作另一首著名的诗歌,这次是写给凯莱肖像基金委员会的。对于那些“被限制在空间”的人来说,他开始说,你能给一个思想已经超越这些界限的人什么样的荣誉呢?然后,在诗意地列举了凯莱的成就之后,他希望观众能在二维肖像前停下来,思考一下这个“灵魂太大,无法容纳普通空间,却在n维空间中自由自在地成长”的人。9很好地唤起了数学家在想象不受我们普通世界束缚的空间时所能感受到的自由感。

Speaking of Cayley, twenty years later, in 1874, he was to be honoured with a portrait; it still hangs in the Trinity College dining room, next to Maxwell’s portrait and across from Newton’s (Maxwell had returned to Cambridge in 1871, as the first director of the university’s first science lab, the Cavendish). The occasion inspired another of Maxwell’s famous poems, this time addressed to the Cayley portrait fund committee. For those who are “to space confined,” he began, what honour could you pay to one whose mind has penetrated beyond these bounds? Then, after poetically listing Cayley’s achievements, he hoped viewers would pause before the portrait’s two-dimensional form and reflect on the man whose “soul, too large for vulgar space, in n-dimensions flourished unrestricted.”9 It’s a lovely evocation of the sense of freedom mathematicians can feel when they imagine spaces untethered to our ordinary world.

不过,凯莱最终研究的是欧几里得几何,而黎曼则研究的是曲面空间或流形的内在几何。用现代术语来说,流形“局部”看起来像平坦的n维欧几里得空间——因此,它是高斯思想的延伸,即如果放大到足够近,曲面看起来是平坦的,而平坦空间可以用学校几何的简单概括来处理,尤其是毕达哥拉斯定理。因此,黎曼通过类比欧几里得度量定义了平坦n维流形的距离测度——度量,或他所谓的“线元素”:

Cayley was ultimately working in Euclidean geometry, though, whereas Riemann was looking at the intrinsic geometry of curved spaces, or manifolds. In modern terminology, a manifold looks “locally” like a flat n-D Euclidean space—so it is an extension of Gauss’s idea that curved surfaces look flat if you zoom in close enough, and that flat spaces can be handled with simple generalisations of school geometry, especially Pythagoras’s theorem. So, Riemann defined the distance measure—the metric, or what he called the “line element”—of a flat n-dimensional manifold by analogy with the Euclidean metric:

ds2=d12+d22+...+dn2

ds2=dx12+dx22+...+dxn2.

事实上,黎曼首先使用“平坦”一词来描述线元素为微分平方和的曲面。

In fact, it is Riemann who first used the term “flat” for a surface whose line element is the sum of squares of differentials like this.

在一张平面纸上,欧氏度量ds 2 = dx 2 + dy 2无论纸张大小如何都成立——而在普通欧氏空间中,ds 2 = dx 2 + dy 2 + dz 2处处成立。然而,在弯曲流形上,黎曼遵循高斯的观点,指出如果你将注意力集中在平面局部,你只能将度量写成像这样的微分和周围的“邻域” 。曲面固有曲率的信息包含在曲面固有度量的微分系数中,正如我们之前在高斯的二维度量中看到的那样,

On a flat piece of paper, the Euclidean metric ds2 = dx2 + dy2 holds regardless of the size of the sheet—and in ordinary Euclidean space, ds2 = dx2 + dy2 + dz2 holds everywhere. On a curved manifold, however, Riemann followed Gauss, noting that you could only write the metric as a sum of differentials like this if you focussed your attention locally, in the flat “neighbourhood” around a point. Information about the intrinsic curvature of the surface is contained in the coefficients of the differentials in the surface’s intrinsic metric, as we saw earlier with Gauss’s 2-D metric,

ds 2 = Edp 2 + 2 Fdpdq + Gdq 2

ds2 = Edp2 + 2Fdpdq + Gdq2.

但一般而言,度量中的系​​数不仅取决于曲率,还取决于坐标的选择:相同的度量在不同的坐标中表示时会有所不同,就像圆的方程在笛卡尔坐标中和在极坐标中看起来不同:对于半径为a且以原点为中心的圆,分别为x 2 + y 2 = a 2r = a。因此,仅仅知道度量中的系​​数并不足以告诉您曲面是否弯曲。然而,黎曼暗示的是,有一种方法可以从这些度量系数中解读流形的曲率。我们稍后会看到他的意思,因为他在这篇针对普通读者的论文中没有详细介绍。

In general, though, the coefficients in a metric depend not just on the curvature but also on the choice of coordinates: the same metric will look different when expressed in different coordinates, just as the equation of a circle looks different in Cartesian coordinates than it does in polar ones: x2 + y2 = a2 and r = a, respectively, for a circle with radius a and centred at the origin. So, the simple fact of having coefficients in the metric is not enough to tell you if the surface is curved. What Riemann hinted at, however, was that there is a way to decipher the manifold’s curvature from these metric coefficients. We’ll see what he meant later, for he didn’t go into detail in this paper, which was designed for a general audience.

1854 年,他在申请成为哥廷根的私人讲师时提出了这个提案。私人讲师的薪水由学生而不是大学支付,因此如果学生不多,申请难度会很大。但这是迈向学术阶梯的第一步,而艰苦的申请过程包括“任教资格”论文和演讲。1908 年,爱因斯坦的第一份学术工作是伯尔尼大学的私人讲师;前一年,他申请以 1905 年的狭义相对论论文作为任教资格论文,但失败了:论文以“难以理解”为由被拒绝!由此可见他的理论在当时有多么激进。黎曼的演讲也超出了大多数听众的理解范围,不过,当时在场的 77 岁高斯肯定意识到了它的意义。高斯是个值得信任的好人,他认为黎曼是一位真正的数学家,“具有极其丰富的独创性”。这的确是极高的赞誉,因为高斯并不是一个喜欢夸赞的人,而格拉斯曼和其他许多人都发现这一点。也许高斯的脾气暴躁与他心爱的第一任妻子难产而死有关,因为他似乎从未从悲痛中恢复过来。10

He presented it when he was applying to become a Privatdozent at Göttingen, in 1854. Privatdozents were lecturers who were paid by their students, not by the university—so it was pretty tough if you didn’t get many students. But it was the first step on the academic ladder—and part of the grueling application process included a “habilitation” thesis and lecture. Einstein’s first academic job, in 1908, was as a Privatdozent at Bern University; he’d failed the year before with an application offering his 1905 special relativity paper as a habilitation thesis: it was rejected as “incomprehensible”! Which shows just how radical his theory was at the time. Riemann’s lecture, too, was beyond most of his listeners’ comprehension, although seventy-seven-year-old Gauss, who was in the audience, certainly appreciated its significance. Gauss was a good person to have on your side, and he thought Riemann was a true mathematician “of a gloriously fertile originality.” High praise indeed, for Gauss wasn’t one to hand out compliments, as Grassmann and many others had found. Perhaps Gauss’s difficult temperament had something to do with the death in childbirth of his beloved first wife, for it seems he never recovered from his grief.10

三年后,黎曼成为哥廷根大学的助理教授,最终成为教授。他为数学,包括他的博士论文,这篇论文为复分析奠定了基础,将复数的概念带入了更深的领域。他还开创了拓扑学的思想,即根据曲面的“属”(现在的称呼)对曲面进行分类——本质上是曲面上“孔”的数量,就像一个只有一个孔的茶杯和一个没有孔的球体。但现在我们的故事关注的是他 1861 年写的一篇论文,因为它是曲率代数理论和张量概念的核心。

Three years later, Riemann became an assistant professor, and eventually a professor, at Göttingen. He made many brilliant contributions to mathematics, including his PhD thesis, which laid the foundations for complex analysis, taking the idea of complex numbers into deeper territory. He also pioneered the topological idea of classifying a surface according to its “genus” (as it is now called)—essentially the number of “holes” in it, like a teacup with one hole as opposed to a sphere with none. But what concerns our story now is a paper he wrote in 1861, for it is at the heart of both the algebraic theory of curvature and the concept of tensors.

黎曼的里程碑式论文:探究张量

RIEMANN’S LANDMARK ESSAY: HOMING IN ON TENSORS

黎曼曾为巴黎科学院主办的一项竞赛撰写了这篇论文,该竞赛的主题是特定类型热分布的热传导。正如我们在格拉斯曼和杰曼的获奖论文中看到的那样,此类竞赛是激发前沿课题研究的重要方式。而且由于它是如此前沿,事情变得有点复杂,尤其是因为黎曼的希腊符号,所以请随意浏览本节以了解关键要点。特别要注意的是,他的希腊符号有多个索引或下标——这是张量的标志——并且度量是“二次微分形式”的例子;此外,符号 Σ 表示和。

Riemann had written this essay for a competition sponsored by the Parisian Academy of Sciences, on the topic of heat conduction for a specific type of heat distribution. As we saw with Grassmann’s and Germain’s prizewinning papers, such competitions were an important way to stimulate research on cutting-edge topics. And since it was so cutting-edge, here things become a little more complex, not least because of Riemann’s Greek notation, so feel free to skim this section for the key take-away points. In particular, note that his Greek symbols have more than one index or subscript—a hallmark of tensors—and that metrics are examples of “quadratic differential forms”; also, the notation Σ means a sum.

黎曼从约瑟夫·傅立叶于 1822 年推导出的热方程开始。正如我在第 6 章中提到的,这个方程显示了热量在物体中扩散时温度如何随时间变化。你可以想象,握住一根金属拨火棍,另一端被火加热,逐渐感觉到热量向上流动并穿过手柄——因此,温度在空间的所有三个维度上都在变化。然而,对于竞赛,学院规定,随着热量的流动,温度应该只在二维上变化——比如,当拨火棍的内部是隔热的,热量只沿着和穿过其表面流动。黎曼对这个问题的创新方法是将热方程中的坐标从通常的笛卡尔坐标xx 1yx 2zx 3转换为新的坐标s 1s 2s 3,他将其定义为仅有二维的函数——比如xy,但不是z。这就是高斯在用pq表示二维曲线度量时所做的。

Riemann began with the heat equation that Joseph Fourier had derived in 1822. As I mentioned in chapter 6, this equation shows how temperature changes over time as heat diffuses through a body. You can imagine, say, holding a metal fire poker where the other end is heated by the fire, and gradually you feel the heat flowing up and through the handle—so the temperature is changing in all three dimensions of space. For the competition, however, the Academy had specified that as the heat flowed, the temperature should change only in two dimensions—say, when the interior of the poker is insulated so the heat only flows along and across its surface. Riemann’s innovative approach to this problem was to transform the coordinates in the heat equation from the usual Cartesian coordinates xx1, yx2, zx3 to new coordinates s1, s2, s3, which he defined to be functions of only two dimensions—say x and y but not z. This is what Gauss had done when he wrote his 2-D curved metric in terms of p and q.

我在这里和第 9 章中讨论了很多关于坐标变换的内容,以及标量和矢量积等量以及欧几里得和闵可夫斯基度量给出的距离/间隔度量在用新坐标表示时如何保持不变。同样,黎曼最终得到了方程

I’ve been talking a lot about coordinate transformations here and in chapter 9, and how quantities such as scalar and vector products, and the distance/interval measures given by the Euclidean and Minkowski metrics, remain invariant when expressed in terms of the new coordinates. Similarly, Riemann ended up with the equation

Σα ι,ι' dx ι dx ι' = Σβ ι,ι' ds ι ds ι'

Σαι,ι′ dxιdxι′ = Σβι,ι′ dsιdsι′.

可以看出等式左边的表达式形式与右边的表达式形式相同,因此该表达式在从x 1 , x 2 , x 3s 1 , s 2 , s 3的黎曼坐标变换下不变。

You can see that the form of the expression on the left-hand side of the equation is the same as that on the right, and so the expression is invariant under Riemann’s coordinate transformation from x1, x2, x3 to s1, s2, s3.

黎曼的二指标符号 α ι,ι′与热方程中的“传导系数”有关,而他的 β ι,ι′与热方程中以新坐标s 1s 2s 3表示的传导系数有关。(他将传导系数本身分别写为a ι,ι′b ι,ι′。)但技术细节11并不重要:重要的是黎曼表示系数的方式,以及他在张量正式定义的四十年前就直觉地知道它们就是我们现在所说的张量的“分量”。

Riemann’s two-index symbol αι,ι′ is related to the “conductivity coefficients,” which appear in the heat equation—and his βι,ι′ is related to the conductivity coefficients in the heat equation written in terms of the new coordinates s1, s2, s3. (He wrote the conductivity coefficients themselves as aι,ι′ and bι,ι′, respectively.) But the technical details11 are not important: it’s the way Riemann represented his coefficients that matters here, and the fact that he intuited—forty years before tensors were formally defined—that they were what we now call the “components” of a tensor.

黎曼方程中的 Σ 代表和。(它是大写的希腊字母 sigma,类似于拉丁文 S,用作“和”的简写。)和 Σα ι,ι′ dx ι dx ι′ 中的指标 ι、ι′分别指代空间的三个维度,由三个坐标x 1x 2x 3表示。在这种情况下,ι、ι′ 的所有可能组合是 (1,1)、(1,2)、(1,3)、(2,1)、(2,2)、(2,3)、(3,1)、(3,2)、(3,3)。因此 Σα ι,ι′ dx ι dx ι′是以下简写方式

The Σ in Riemann’s equation stands for sum. (It’s the upper-case Greek letter sigma, analogous to the Latin S, used as shorthand for “sum.”) The indices ι, ι′ in the sum Σαι,ι′dxιdxι′ each refer to the three dimensions of space, represented by the three coordinates x1, x2, x3. All the possible combinations of ι, ι′ in this case are (1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3). So Σαι,ι′dxιdxι′ is a shorthand way of writing

α 1,1 dx 1 dx 1 + α 1,2 dx 1 dx 2 + α 1,3 dx 1 dx 3 + α 2,1 dx 2 dx 1 + … + α 3,3 dx 3 dx 3 .

α1,1dx1dx1 + α1,2dx1dx2 + α1,3dx1dx3 + α2,1dx2dx1 + … + α3,3dx3dx3.

(Σβ ι,ι′也类似。)这看起来很像我们一直在讨论的度量,尽管这种“微分形式”也纯粹是为了其代数性质而被研究的——而黎曼谈论的是热传导,而不是度量。微分形式是带有微分的表达式,例如dx 1dx 2 ,... ;黎曼的表达式是“二次微分形式”,因为它由两个“微分”的乘积组成—— dx 1 dx 1dx 1 dx 2,等等——类似于二次方程中的平方。

(And similarly for Σβι,ι′.) This looks rather like the metrics we’ve been discussing, although this kind of “differential form” had also been studied purely for its algebraic properties—and Riemann was talking about heat conduction, not metrics. Differential forms are expressions with differentials such as dx1, dx2, … ; Riemann’s expression is a “quadratic differential form,” because it is made from products of two “differentials”—dx1dx1, dx1dx2, and so on—analogous to the squares in quadratic equations.

为了看出黎曼微分形式与度量的相似性,在欧几里得度量中d12+d22+d32与黎曼的 α ι,ι′类似的系数为:

To see the similarity of Riemann’s differential form with a metric, in the Euclidean metric dx12+dx22+dx32 the coefficients analogous to Riemann’s αι,ι′ would be:

α11 =α22 = α33 = 1

α11 = α22 = α33 = 1,

其余 α ij都等于零。(今天,张量分量中的指标不再像黎曼那样用逗号分隔,因为逗号现在指的是偏导数。我使用下标ij是因为黎曼对 ι 和 ι′ 的使用令人困惑:如图9.3所示,破折号通常表示变换后的坐标。但在本章的其余部分,我将保留黎曼的符号。)黎曼非常清楚这种与度量的相似性。不过,尽管他说他使用的方法与高斯在曲面上工作时使用的方法类似,但他的重点是从热方程分析中得出的代数。

with all the other αij’s equal to zero. (Today the indices in a tensor component aren’t separated by commas as Riemann did, because commas now refer to partial derivatives. And I’ve used the subscript ij because Riemann’s use of ι and ι′ is confusing: as we’ve seen in figure 9.3, dashes often denote transformed coordinates. But I’ll keep to Riemann’s notation for the rest of this chapter.) Riemann was well aware of this similarity with metrics. But although he said that he was using similar methods to those Gauss had used for his work on curved surfaces, his focus was on the algebra that came out of his analysis of the heat equation.

第一个暗示表明,这种代数分析涉及到我们现在称之为张量的量,是他用来表示热方程中的电导率系数( a ι,ι′b ι,ι′ )以及我们在上面二次形式中看到的相关系数 α ι,ι′和 β ι,ι′ 的二指标符号。在第 9 章末尾,我展示了麦克斯韦在用两个指标P hk表示应力分量时,也是直观地想到了张量的概念。我还借助图 9.49.5展示了为什么他需要两个指标,而不是像矢量分量那样需要一个指标(例如如图 9.1所示)。黎曼并没有讨论为什么在将电导率系数表示为a ι,ι′时使用两个指标。但为了直观地展示它们,你可以取一小块导热体,如图 9.4所示——因此系数a ι,ι′ 的行为与P hk类似,只是它们测量的是导热性而不是应力。例如,当 ι ≡ y和 ι′ ≡ x时,热传导沿y - x方向进行,如箭头P yx所示。

The first hint that this algebraic analysis involved quantities that we now call tensors is in the two-index notation he used to represent the conductivity coefficients (aι,ι′ and bι,ι′) in the heat equation, and the associated coefficients αι,ι′ and βι,ι′ that we saw in the quadratic form above. At the end of chapter 9, I showed how Maxwell, too, had intuited the idea of a tensor, when he wrote the components of stress with two indices, Phk. I also showed—with the help of figures 9.4 and 9.5—why he needed two indices, rather than the one you need for the components of a vector (as in fig. 9.1, for example). Riemann didn’t discuss why he used two indices when he represented his conductivity coefficients as aι,ι′. But to visualise them, you can take a small volume of the body conducting the heat, just as in figure 9.4—so the coefficients aι,ι′ behave like the Phk, except that they are measuring conductivity rather than stress. For example, when ι ≡ y and ι′ ≡ x, the conduction of heat is going in the y-x direction like the arrow Pyx.

黎曼确实指出,他只考虑了传导在两个方向上相同的情况——例如,当传导从yx与从xy相同时,即a y , x = a x , y,或者更一般地,a ι,ι′ = a ι′,ι 。这是不变性的另一个例子:交换a ι,ι′上的指标不会改变系数的值。另一个名字因为这种不变性就是“对称性”,因为交换指标 ι、ι′ 得到 ι′、ι 就像反射它们一样,就像在镜子中反射你的图像,或者绕对称轴反射雪花一样。

Riemann did specify that he was considering only the case where the conduction was the same in both directions—for example, when the conduction is the same from y to x as from x to y, so that ay,x = ax,y, or more generally, aι,ι′ = aι′,ι. This is another example of invariance: interchanging the indices on aι,ι′ doesn’t change the value of the coefficient. Another name for this invariance is “symmetry,” because interchanging the indices ι, ι′ to get ι′, ι is like reflecting them, just like reflecting your image in a mirror, or reflecting a snowflake about an axis of symmetry.

但黎曼的“张量”不仅仅只有两个指标。

But there’s more to Riemann’s “tensors” than just two indices.

张量不仅仅是符号!

THERE’S MORE TO TENSORS THAN NOTATION!

事实上,仅用两个指标不一定能代表张量。例如,在黎曼论文发表 20 年后,麦克斯韦也用两个指标来表示电导率——他用的是K pq,他解释说电导是从点p流向q的;但他也用C pq来表示流向同一方向的电流。12电流确实有方向和大小,但在电路中它像普通数字或标量一样相加,所以它既不是矢量也不是张量。

In fact, two indices alone are not necessarily the marker of a tensor. For instance, two decades after Riemann’s paper, Maxwell, too, represented conductivity with two indices—in his case, Kpq, where he explained that the conduction was flowing from a point p to q; but he also denoted a current flowing in the same direction as Cpq.12 It is true that a current has a direction and a magnitude, but in a circuit it adds like an ordinary number or scalar, so it is neither a vector nor a tensor.

符号问题重要,而指数符号对于用张量进行计算非常有用。但它只是表示数学量的一种方式,而不是定义数学量的方式。重要的是数学量如果要被称为向量和张量所必须具备的属性——不仅包括它们的加法和乘法规则,还包括表达不变表达式的能力,比如标量积ab和黎曼的 Σα ι,ι′ dx ι dx ι′ = Σβ ι,ι′ ds ι ds ι′。稍后我们将更详细地看到定义张量需要什么。但事实证明,黎曼的 α ι,ι′和 β ι,ι′是张量分量——电导率系数也是如此,尽管黎曼的计算没有显示这一点。13毕竟,张量还没有被发明!尽管如此,黎曼的论文——就像麦克斯韦、柯西和汤姆森关于应力的论文一样——展示了导致发明它们成为必要的问题类型。

The question of notation is important, and the index notation is brilliant for doing computations with tensors. But it is only a way of representing a mathematical quantity, not of defining it. What matters is the properties that mathematical quantities have to have if they are to be called vectors and tensors—including not just their rules of addition and multiplication but also the ability to express invariant expressions, such as the scalar product ab and Riemann’s Σαι,ι′dxιdxι′ = Σβι,ι′dsιdsι′. Later we’ll see in more detail what it takes to define a tensor. But it does turn out that Riemann’s αι,ι′ and βι,ι′ are tensor components—and so are the conductivity coefficients, although Riemann’s working doesn’t show it.13 After all, tensors haven’t yet been invented! Still, Riemann’s paper—like Maxwell’s, Cauchy’s, and Thomson’s papers on stress—shows the kinds of problems that made it necessary to invent them.

不过,张量可以有两个以上的指标,在​​寻找使 Σβ ι,ι′ ds ι ds ι′保持不变(即与 Σα ι,ι′ dx ι dx ι′相同)的变换时,黎曼也提出了三和四指标量。然后他做了一些非常特别的事情。首先,他抛出了这样一个事实:“表达式βιιdsιdsι可以看作是更一般的n维空间中的线元素,超出了我们直觉的范围。”然后他说,如果你想象这个空间中的一个表面,那么他的三和四指标量就是测量的关键曲面的曲率。它们由系数 β ι,ι′及其导数的组合构成——所以这就是黎曼在他的任教演讲中所说的意思,当时他建议有一种方法可以从度量系数中梳理出曲率信息,而不管坐标系如何。14

There’s more, though, for tensors can have more than two indices, and in his search for the transformation that kept Σβι,ι′dsιdsι′ invariant—that is, the same as Σαι,ι′dxιdxι′—Riemann came up with threeand four-index quantities, too. Then he did something really special. First, he tossed off, as an aside, the fact that “the expression βι,ι,dsιdsι, can be regarded as a line element in a more general space of n dimensions extending beyond the bounds of our intuition.” Then he said that if you imagine a surface in this space, then his three- and four-index quantities are key to measuring the curvature of the surface. They are built from combinations of the coefficients βι,ι′ and their derivatives—so this is what Riemann meant in his habilitation lecture, when he suggested there was a way to tease out the curvature information from the metric coefficients, regardless of the coordinate system.14

黎曼没有给这些三指标和四指标量命名。它们本质上是现在(感谢里奇和列维-奇维塔)分别称为克里斯托费尔符号和黎曼张量的分量。“克里斯托费尔符号”这个名称表明这些三指标量不是张量的分量——至少不是单个张量的分量。这里的细节并不重要,因为要点再次简单:如果某物有指标,它不一定是张量;正如我所指出的,它需要具有其他属性——尤其是线性坐标变换下的不变性。

Riemann didn’t give a name to these three- and four-index quantities. They are essentially the components of what are now—thanks to Ricci and Levi-Civita—called the Christoffel symbols and the Riemann tensor, respectively. The name “Christoffel symbols” indicates that these three-index quantities aren’t components of a tensor—at least, not of a single tensor. The details don’t matter here, for the point once again is simply that if something has indices, it isn’t necessarily a tensor; as I indicated, it needs to have other properties—notably invariance under linear coordinate transformations.

克里斯托费尔符号由二次微分形式(如度量)中系数的导数构成,而黎曼张量则由克里斯托费尔符号及其导数构成。因此,在具有度量的空间中,关键思想是,如果黎曼张量等于零,则空间是平坦的。此外,如果空间是平坦的,则黎曼张量为零。这意味着黎曼张量可以告诉你你的空间是弯曲的还是平坦的。所以它通常被称为“曲率张量”。

Christoffel symbols are built from derivatives of the coefficients in a quadratic differential form such as the metric, and the Riemann tensor is built from the Christoffel symbols and their derivatives. So, in a space with a metric, the key idea is that if the Riemann tensor equals zero, then the space is flat. What’s more, if the space is flat, then the Riemann tensor is zero. Which means the Riemann tensor is the thing that tells you if your space is curved or flat. So it’s often just called the “curvature tensor.”

由于黎曼张量是根据度量中的系​​数构建的,而度量中的系​​数是坐标的函数,因此它在空间中的每个点都有一个值。因此从技术上讲,它是一个张量场而不是张量,就像电矢量和磁矢量EB是矢量场一样。但在实践中,研究人员往往简单地指定义引力场和电磁场的张量矢量。(然而,有些人要求更精确的数学语言,但这种严格的定义是在麦克斯韦对法拉第物理场思想的数学发展和黎曼对曲率的开创性工作之后很久才出现的。)

Since the Riemann tensor is built from the coefficients in the metric, which are functions of the coordinates, it has a value at each point in space. So technically it is a tensor field rather than a tensor, just as the electric and magnetic vectors E and B are vector fields. In practice, though, researchers tend to refer simply to the tensors and vectors that define gravitational and electromagnetic fields. (Some, however, demand more mathematically precise language, but such rigorous definitions came long after Maxwell’s mathematical development of Faraday’s idea of physical fields, and Riemann’s pioneering work on curvature.)

黎曼张量有时被称为黎曼-克里斯托费尔张量,组成它的符号被称为克里斯托费尔符号,因为令人惊讶的是,黎曼并不是唯一一个提出这些量的人。爱因斯坦老学校的数学教授埃尔温·克里斯托费尔也提出了这些量。克里斯托弗的母校是苏黎世理工大学,尽管他在 1869 年发表了他的论文,比爱因斯坦在那里读书早了三十年。克里斯托弗可能不知道黎曼 1861 年的论文,但他受到了黎曼 1854 年任教资格讲座的启发,这促使他探索二次微分形式不变性的条件。然而,与黎曼不同,克里斯托弗没有将他的三和四指标符号与曲率联系起来,因为他的重点是纯代数。15

The Riemann tensor is sometimes called the Riemann-Christoffel tensor, and the symbols comprising it are called the Christoffel symbols because, surprisingly, Riemann wasn’t the only one to come up with these quantities. So did Elwin Christoffel, a maths professor at Einstein’s old school, the Zurich Polytechnic—although Christoffel published his work in 1869, three decades before Einstein was a student there. Christoffel probably didn’t know about Riemann’s 1861 essay, but he was inspired by Riemann’s 1854 habilitation lecture, which led him to explore conditions for the invariance of quadratic differential forms. Unlike Riemann, though, Christoffel didn’t connect his three- and four-index symbols with curvature, for his focus was pure algebra.15

黎曼没有进一步发展他的曲率思想——这要由里奇、爱因斯坦和格罗斯曼来决定。就像他的英文翻译克利福德——以及麦克斯韦、闵可夫斯基和其他许多人一样——黎曼死得太早,无法充分发挥他的潜力。1862 年,他感染了肺结核。在接下来的几年里,他和新婚妻子以及小女儿在意大利度过了一段时间,迫切希望他能在温暖的气候中康复。但在 1866 年,他与这种可怕的疾病抗争失败了,就像他的母亲和三个姐妹一样。那时他还不到四十岁。

Riemann didn’t develop his curvature idea much further—that would be up to Ricci, Einstein, and Grossmann. For like his English translator Clifford—and like Maxwell, Minkowski, and so many others—Riemann died too young to fully develop his potential. In 1862, he’d contracted tuberculosis. Over the next few years, he and his new wife and baby daughter spent time in Italy, desperately hoping he’d recuperate in the warmer climate. But in 1866, he lost his battle with this dreadful disease, just like his mother and three sisters. He wasn’t yet forty.

• • •

• • •

黎曼的任教资格演讲稿直到 1867 年才发表。克利福德的英文版《论几何基础假设》发表于 1873 年的《自然》杂志上。黎曼关于热的论文只在他 1876 年的文集中发表——它没有赢得论文比赛!评委们在寻找更具体的东西——他们并没有欣赏黎曼对热问题的异常普遍的方法,更不用说它包含了张量分析和曲率理论的种子。所以,他的论文多年来一直无人问津,无人知晓,黎曼从来不知道它会变得如此重要。他从来不知道他的名字会永远留在黎曼张量这个广义相对论的基石上。

Riemann’s habilitation lecture wasn’t published until 1867. Clifford’s English version, On the Hypotheses Which Lie at the Bases of Geometry, appeared in 1873, in Nature. Riemann’s essay on heat was published only in his collected works of 1876—it hadn’t won the essay competition! The judges were looking for something more specific—they hadn’t appreciated Riemann’s extraordinarily general approach to their heat problem, let alone the fact that it contained the seeds of both tensor analysis and the theory of curvature. So, his paper had languished for years, unappreciated and unknown, and Riemann never knew how important it would become. He never knew that his name would live on in the Riemann tensor, the bedrock of general relativity.

然而,1912 年,爱因斯坦和格罗斯曼利用黎曼的工作建立了弯曲时空的几何学。我们之前看到,爱因斯坦已经确定了他和格罗斯曼需要解决的两个问题:如何将物理定律从狭义理论转移到广义理论;以及如何在弯曲时空中找到适当的度量。黎曼给了他们第二个问题的答案:线元素将看起来像黎曼的βιιdsιdsι系数 β ι,ι′将是坐标的函数。它们不会是常数,就像欧几里得度量中的 1,或闵可夫斯基度量中的1, 1, 1, − c 2 ,

In 1912, however, Einstein and Grossmann seized upon Riemann’s work in order to build the geometry of curved space-time. We saw earlier that Einstein had identified two problems he and Grossmann needed to solve: how to transfer the laws of physics from the special theory to the general one; and how to find the appropriate metric in curved space-time. Riemann gave them the answer to the second problem: the line element will look like Riemann’s βι,ι,dsιdsι, and the coefficients βι,ι′ will be functions of the coordinates. They won’t be constants, like the 1s in the Euclidean metric, or the 1, 1, 1, −c2 in Minkowski’s metric,

ds 2 = dx 2 + dy 2 + dz 2c 2 dt 2

ds2 = dx2 + dy2 + dz2c2 dt2,

因为那时它们的导数将为零,因此黎曼张量也将为零,所以时空将是平坦的。16

for then their derivatives would be zero and therefore so would the Riemann tensor, so the space-time would be flat.16

然而,要解决第一个问题,爱因斯坦和格罗斯曼需要一个严谨的理论,该理论将我讨论过的所有张量思想结合起来——这些思想是许多数学家几十年来凭直觉得出的,从柯西和克里斯托弗到麦克斯韦、黎曼等等。所以,是时候见见格雷戈里奥·里奇了!

To solve the first problem, however, Einstein and Grossmann will need a rigorous theory incorporating all the tensorial ideas I’ve been discussing—ideas intuited over many decades by many mathematicians, from Cauchy and Christoffel to Maxwell, Riemann, and more. So, it’s time to meet Gregorio Ricci!

(11)张量的发明及其重要性

(11) INVENTING TENSORS—AND WHY THEY MATTER

1861 年,当格雷戈里奥·里奇八岁时,一系列政治和军事阴谋最终导致意大利王国的成立。这意味着,大多数分散的意大利州、公国和王国最终正式统一起来,尽管并不总是心甘情愿。但宗教主导了里奇的童年。他的父亲是一位贵族地主、商人和工程师,是一位虔诚的罗马天主教徒,他不仅通过向教堂捐赠巨额款项来表达自己的信仰,还通过救济饥饿的人来表达自己的信仰。他的母亲会带着四个孩子走上街头寻找穷人,尤其是妇女。她似乎扮演着顾问的角色,倾听妇女们的烦恼并提供安慰。小里奇抱怨母亲不时停下来等待,听她讲述每一个悲惨的故事,但她在这种情况下的尊严给他留下了深刻的印象,他也是一名终生虔诚的天主教徒。1

In 1861, when Gregorio Ricci was eight years old, a long series of political and military machinations culminated in the proclamation of the Kingdom of Italy. This meant that most of the disparate Italian states, duchies, and kingdoms were formally, if not always willingly, united at last. But it was religion that dominated Ricci’s childhood. His father—an aristocratic landowner, businessman, and engineer—was a devout Roman Catholic, who expressed his faith not just through hefty donations to the church but also by feeding the hungry. And his mother would take her four children with her as she walked the streets seeking out the poor—especially the women. She seemed to act as a counselor, hearing the women’s troubles and offering comfort. Young Ricci grumbled at all the periodic stopping and waiting while his mother listened to each tale of woe, yet her dignity in the situation left a lasting impression on him, and he, too, was a lifelong and devout Catholic.1

顺便说一句,利玛窦的姓氏是利玛窦·库尔巴斯特罗 (Ricci Curbastro),但在他关于张量微积分的里程碑式论文中,他署名为利玛窦,所以这就是他今天为人所知的名字。(我应该提到,同样,詹姆斯·克拉克·麦克斯韦的姓氏是克拉克·麦克斯韦,但他的朋友都称他为麦克斯韦。)年轻的利玛窦是一个热心的学生,用他的一位老师的话来说,他有着不同寻常的“敏锐的头脑”和“活泼的创造力”。2 1869年,他开始在罗马的教皇大学学习数学——但在 1870 年夏天,他不得不因为另一场战争而回国。法国人与教皇军队结盟,当普鲁士人占上风时,意大利的新国王抓住机会接管了罗马,并将其纳入新的意大利王国。

Incidentally, Ricci’s family name was Ricci Curbastro, but in his landmark paper on tensor calculus he signed himself Ricci, so that’s the name he’s known by today. (I should mention that similarly, James Clerk Maxwell’s family name was Clerk Maxwell, but his friends referred to him as Maxwell.) Young Ricci was a keen student, with an unusually “penetrating mind” and a “lively ingenuity,” as one of his teachers put it.2 Then in 1869 he began to study mathematics at the papal university in Rome—but in the summer of 1870, he had to return home because of yet another war. The French had been allied with the papal troops, and when the Prussians prevailed, the new king of Italy seized the chance to take over Rome and bring it into the new Italian kingdom.

罗马现在是新国家的世俗首都,前教皇大学建筑被国家接管,利玛窦将目光投向了历史悠久的博洛尼亚大学,该大学离他位于意大利东北部的家乡卢戈·迪·罗马涅更近。他必须再学习两年才能获得入学资格,因此所有这些政治动乱让他付出了高昂的代价。尽管如此,他仍然是统一的支持者,他告诉一位朋友,在博洛尼亚也意味着“我可以更加专注和热情地关注我们的同胞在卢戈将要实施的政治改革,他们创造了一个统一的意大利。” 3

With Rome now the secular capital of the new nation, and the former papal university building taken over by the state, Ricci set his sights on the venerable University of Bologna, which was much closer to his hometown of Lugo di Romagno in northeast Italy. He had to do some extra study to qualify for admission—two years’ worth, so all that political upheaval cost him dearly. Still, he was a supporter of the unification, and being at Bologna also meant, he told a friend, “I can follow more attentively and with more zeal the political reforms that our countrymen who have created a united Italy are going to carry out [in Lugo].”3

在博洛尼亚度过了辉煌的一年后——他在微积分、化学和几何考试中都取得了满分——他决定再次搬家,这次他搬到了比萨,因为那里的数学学院充满活力。他在 1875 年获得博士学位,第二年获得教师资格证——但当时他找不到工作,因为大学职位很少。所以,他留在比萨,成为一名独立学者,阅读最新的数学和物理学。和爱因斯坦一样,当他发现麦克斯韦的电磁理论时,他特别兴奋——在 1877 年,这确实是你在学校学不到的前沿研究。麦克斯韦本人还活着,而海因里希·赫兹发出无线电波,如此惊人地证实了这一理论,还有十年的时间。

After a stellar year at Bologna—he scored full marks for his exams in calculus, chemistry, and geometry—he decided to move once again, this time to Pisa for its vibrant mathematical school. He got his doctorate in 1875, and his teaching certificate the following year—not that he could get a job at the time, for university positions were scarce. So, he stayed on at Pisa as an independent scholar, reading up on the latest maths and physics. Like Einstein, he was particularly excited when he discovered Maxwell’s theory of electromagnetism—and in 1877, this really was cutting-edge research you couldn’t learn in school. Maxwell himself was still alive, and there was still a decade to go before Heinrich Hertz generated the radio waves that so spectacularly confirmed the theory.

在几次申请教职失败后,里奇获得了公共教育部的奖学金——这就是他师从著名的费利克斯·克莱因。当时在慕尼黑的克莱因因发现不变理论和坐标变换群之间的关系而闻名——三十年后,亨利·庞加莱和爱因斯坦证明了洛伦兹变换形成了一个群。它们是坐标变化,在坐标变化下,闵可夫斯基度量和麦克斯韦方程保持不变,正如我们前面所见——我在图 9.3的方框标题中简要解释了它们为什么形成一个群。克莱因还发展了阿瑟·凯莱关于曲面投影的工作,总而言之,他的数学兴趣如此广泛,以至于当里奇于 1878 年秋天抵达慕尼黑时,克莱因为他提供的学习和研究计划让他应接不暇。尽管如此,他还是认为克莱因是一位“善良”的导师,在学习上给了他“大力帮助” 。4

After a couple of unsuccessful applications for teaching positions, Ricci obtained a fellowship from the ministry of public instruction—which is how he came to study under the famous Felix Klein. Klein, who was then based in Munich, was already celebrated for finding the relationship between invariant theory and groups of coordinate transformations—and three decades later, Henri Poincaré and Einstein would show that the Lorentz transformations form a group. They’re the coordinate changes under which the Minkowski metric and Maxwell’s equations remain invariant, as we saw earlier—and I briefly explained why they form a group in the boxed caption to figure 9.3. Klein also developed Arthur Cayley’s work on projections of curved surfaces, and all in all he had such wide-ranging mathematical interests that when Ricci arrived in Munich in the autumn of 1878, he was overwhelmed by the study and research program Klein offered him. Still, he found Klein to be a “kind” advisor who gave him “vigorous help” in his studies.4

更重要的是,与克莱因共事帮助里奇对自己的能力产生了信心——这是老师送给学生的绝妙礼物。19 世纪 90 年代,奇泽姆在克莱因的指导下攻读博士学位时,她不仅被克莱因的聪明才智所震撼,还被他鼓励学生要有“永不迟钝”的信心所震撼。(1895 年,里奇泽姆获得博士学位后不久——这是德国第一个授予女性的正式博士学位——她嫁给了格顿学院的前导师威廉·杨,所以她今天更广为人知的名字是格蕾丝·奇泽姆·杨。她的博士论文是关于代数群(克莱因的专业)在球面几何中的应用。)5

Even more important, working with Klein helped Ricci develop confidence in his own ability—a wonderful gift from a teacher to a student. When Chisholm did her PhD with Klein in the 1890s, she, too, would be struck not just by his brilliant mind, but also by the way he would encourage his students to have the confidence to “never be dull!” (Not long after she took her doctorate in 1895—the first official doctorate awarded to a woman in Germany—she married her former Girton tutor William Young, so she is better known today as Grace Chisholm Young. Her doctoral thesis was on algebraic groups, Klein’s specialty, applied to spherical geometry.)5

在克莱因工作一年后,利玛窦又花了几年时间寻找大学全职职位,但徒劳无功——直到 1880 年冬天,他终于被任命为帕多瓦大学数学物理学副教授。伽利略、哥白尼和卡尔达诺只是利玛窦在这所古老而进步的大学中杰出的前辈中的三位,而利玛窦将在那里待上四十五年。

After his year with Klein, Ricci spent a couple more years fruitlessly seeking a full-time university position—until finally, in the winter of 1880, he was appointed associate professor of mathematical physics at the University of Padua. Galileo, Copernicus, and Cardano were just three of his illustrious predecessors at this ancient and progressive institution, and Ricci would remain there for the next forty-five years.

作为帕多瓦学者,他发表的第一篇论文是关于电磁学和微分方程的,但他很快就对不变性数学产生了兴趣。他还决定是时候找个妻子了。他在比萨读书时第一次坠入爱河,但女孩对他尴尬的迷恋不屑一顾。后来,他哥哥“不合适的”爱情引起了他的注意里奇的父母是天主教徒,他非常愤怒——她来自一个贫穷且不传统的家庭,但真正让里​​奇感到不安的是,她是他的亲戚,他确信上帝不会同意这种乱伦的结合。因此,30 岁的里奇决定,避免心痛和父母不悦的最好方法是向当地牧师寻求婚介建议。牧师很乐意帮忙,他安排里奇认识了一位活泼、聪明、非常合适的年轻女子比安卡。这是一次幸福的求爱,她成为了他的新娘和终身伴侣。6

His first published papers as a Padua academic were on electromagnetism and differential equations, but he soon became intrigued by the maths of invariance. He also decided it was time to find a wife. He’d first fallen in love when he was a student at Pisa, but the girl had disdained his awkward infatuation. Then his older brother’s “unsuitable” love interest had aroused fury from his conservative Catholic parents—she came from a poor and unconventional family, but what really upset Ricci senior was that she was a relative, and he was sure God would not approve of such an incestuous union. So, thirty-year-old Ricci decided the best way to avoid both heartache and parental displeasure was to seek matchmaking advice from the local priest. The priest was happy to oblige, and he arranged for Ricci to meet a lively, intelligent, and eminently suitable young woman named Bianca. It was a happy courtship, and she became his bride and lifelong companion.6

不过,里奇所做的不仅仅是追求。1884 年,也就是他结婚的那一年,他发表了第一篇关于张量分析之路的论文。他受到了黎曼和高斯作品的启发,他的论文是关于“二次微分形式”的变换性质——即微分对的平方和或其他乘积,比如我们在基于毕达哥拉斯定理的距离度量中看到的。我们在高斯度量中看到了一些这样的性质,它在弯曲变换下是不变的——就像卷起的纸片显示圆柱体的表面本质上和展开的纸一样平坦——以及闵可夫斯基度量的洛伦兹不变性。1876 年,黎曼发表了 1861 年关于热的论文,将高斯的工作扩展到n维,此后,一些数学家接受了他的想法。但里奇的目的是“避免对三维以上空间的存在和性质进行漫无目的的讨论”,并超越同事对微分形式应用的关注相反,他想提供对数学理论的清晰理解。7

Ricci had been doing more than courting, though. In the same year as his wedding, 1884, he published his first paper on the road to tensor analysis. He’d been inspired by the works of Riemann and Gauss, and his paper was on the transformation properties of “quadratic differential forms”— that is, sums of the squares or other products of pairs of differentials, such as we’ve seen in the distance metrics based on Pythagoras’s theorem. We saw some of these properties with Gauss’s metric, which is invariant under bending transformations—like the rolled-up piece of paper that shows the surface of a cylinder is just as intrinsically flat as the unrolled paper—and with the Lorentz-invariance of Minkowski’s metric. After the 1876 publication of Riemann’s 1861 essay on heat, which extended Gauss’s work to n-dimensions, several mathematicians had taken up his ideas. But Ricci’s aim was to “avoid lazy discussions about the existence and nature of spaces of more than three dimensions,” and to look beyond his colleagues’ focus on applications of differential forms. Rather, he wanted to provide a clear understanding of the mathematical theory.7

四十年前,汉密尔顿对向量代数和向量微积分规则的开发,让我们看到了这种对理论的重视。他从i、j、k 的乘法规则开始,这些规则是他在布鲁姆桥上成功雕刻出来的,这些规则使他能够定义新的乘法——四元数、标量和向量积。掌握了这些之后,他创造了微分微积分向量算子 nabla,∇。直到麦克斯韦从泰特那里学到了所有这些规则——并给 nabla 运算命名为 grad、divergence 和 curl——他才觉得在他的电磁学理论中,他已经熟练地使用矢量微积分。所以现在,为张量建立类似的规则正是里奇想要做的事情。

We saw this emphasis on theory with Hamilton’s development of the rules of vector algebra and vector calculus, forty years earlier. He’d begun with the rules for multiplying his i, j, k, which he’d triumphantly carved on Broome Bridge, and they enabled him to define new kinds of multiplication—quaternion, scalar, and vector products. Once he had these in hand, he’d created the differential calculus vector operator nabla, ∇. It was only after Maxwell had learned all these rules from Tait—and had given names to the nabla operations grad, divergence, and curl—that he’d felt comfortable using vector calculus in his theory of electromagnetism. So now, establishing similar rules for tensors is just what Ricci set out to do.

他并没有把它们称为张量,而是简单地将它们称为“函数系统”。例如,黎曼的“线元素”(或度量)Σβ ι,ι′ ds ι ds ι′中的系数 β ι, ι′现在被称为张量的分量,而且,正如我们在第 10 章中看到的,在曲面上,它们是坐标的函数——所以,实际上,它们是函数集(或“系统”!)。(所以从技术上讲,它们是“张量场”,但正如我在上一章中提到的,不太严格地说,“张量”也很好。)我们稍后会从数学上看到为什么度量是张量——以及黎曼的四指标量,里奇称之为“黎曼系统”,所以它现在被称为黎曼张量。里奇不再说“张量微积分”,而是将他的系统的微积分称为“绝对微分学”。他所说的“绝对”是指不变,因为张量的有趣之处在于它们编码了不变性的概念。

Not that he called them tensors: rather, he simply referred to them as “systems of functions.” For example, the coefficients βι,ι′ in Riemann’s “line element” (or metric) Σβι,ι′dsιdsι′ are now called the components of a tensor, and, as we saw in chapter 10, on curved surfaces they are functions of the coordinates—so, in fact, they’re a set (or “system”!) of functions. (So technically they are “tensor fields,” but as I mentioned in the previous chapter, less rigorously “tensors” will do nicely.) We’ll see mathematically why the metric is a tensor later—and Riemann’s four-index quantity, too, which Ricci called “the system of Riemann,” so it’s now called the Riemann tensor. And instead of saying “tensor calculus,” Ricci named the calculus of his systems the “absolute differential calculus.” By “absolute” he meant unchanging, because the interesting thing about tensors is that they encode the idea of invariance.

不过,在深入探讨里奇数学之前,我将概述张量的基本概念,它是一种表示和计算信息的方式——就像向量一样。但首先我要告诉你张量的现代名称是如何得来的。

Before I get into Ricci’s maths, though, I’ll sketch out the basic idea of tensors as a way of representing, and calculating with, information—just like vectors. But first I’ll tell you how tensors got their modern name.

张量的名称由来以及它们如何表示数据

HOW TENSORS GOT THEIR NAME— AND HOW THEY REPRESENT DATA

“张量”这个术语是汉密尔顿通过哥廷根数学教授沃尔德马尔·福格特传给我们的。但真正确立这个名称的是爱因斯坦,当时他和马塞尔·格罗斯曼理解了里奇的绝对微积分。汉密尔顿曾在不同的语境中使用过这个术语——四元数的量,他通过类比复数的模来定义四元数。它是四元数分量平方和的平方根——就像矢量的量可以通过分量的平方和来求得一样。但福格特是第一个在现代语境中使用“张量”这个术语的人,在他 1898 年出版的关于晶体学的书中——顺便说一句,奇泽姆·杨和她的丈夫在《自然》杂志上对这本书给予好评。8

The term “tensor” comes to us from Hamilton via Göttingen maths professor Woldemar Voigt. But it is Einstein who would firmly establish this name, once he and Marcel Grossmann got their heads around Ricci’s absolute calculus. Hamilton had used the term in a different context—as the magnitude of a quaternion, which he defined by analogy with the modulus of a complex number. It’s the square root of the sum of the squares of the components of the quaternion—just as the magnitude of a vector is found from the sum of the squares of its components. But Voigt is the one who first used the term “tensor” in its modern context, in his 1898 book on crystallography—which, incidentally, Chisholm Young and her husband favourably reviewed in Nature.8

Voigt 特别提到了晶体中的应力和张力——“tension” 源自拉丁语“tensio”,而“tensio”又源自“tendere”,意为“拉伸”。但沃格特说,他只是扩展了汉密尔顿对“张量”的使用。例如,矢量或叉积a × b产生第三个矢量c ,其幅值(或汉密尔顿“张量”)与ab的幅值(“张量”)相关。换句话说,叉积从另外两个幅值得出一个新的幅值(“张量”)——正如我们即将看到的,里奇张量分析的新颖之处之一是,张量乘法会从旧张量产生新的“高阶”张量。

Voigt was referring specifically to the stresses and tensions in crystals—and “tension” comes from the Latin “tensio,” which in turn comes from “tendere,” meaning “to stretch.” But Voigt said he was simply extending Hamilton’s use of “tensor.” For example, the vector or cross product, a × b, produces a third vector c whose magnitude (or Hamiltonian “tensor”) is related to the magnitudes (“tensors”) of a and b. In other words, the cross product gives a new magnitude (“tensor”) from two others—and as we’re about to see, one of the novel things about Ricci’s tensor analysis is that tensor multiplication produces new, “higher-order” tensors from old ones.

格拉斯曼也有过这种想法。正如我们在第 5 章中看到的那样,他没有使用术语“矢量”,而是使用了德语单词strecke,可以翻译为“线或拉伸”——他的基本几何对象是可以“拉伸”或“延伸”形成平面的“线”,就像平行四边形规则中两个矢量形成一个平面一样。格拉斯曼将两个三维矢量的外积定义为两个矢量所围成的平行四边形的有向面积。通常的矢量或叉积实际上也是这样做的,但仅限于平行四边形,而格拉斯曼对外积的定义意味着你可以添加第三个矢量来将平行四边形扩展为一个盒子,依此类推,添加任意数量的新维度。

Grassmann had had this idea, too. As we saw in chapter 5, instead of the term “vector,” he’d used the German word strecke, which can be translated as “line or stretch”—and his basic geometric objects were “lines” that could be “stretched” or “extended” to form planes, just as two vectors form a plane in the parallelogram rule. Grassmann defined the outer product of two 3-D vectors as the oriented area of the parallelogram bounded by the two vectors. The usual vector or cross product does this, too, in effect, but only for parallelograms, whereas Grassmann’s definition of an outer product implied that you could then add a third vector to extend the parallelogram to a box, and so on, adding as many new dimensions as you like.

19 世纪 80 年代,吉布斯进一步发展了格拉斯曼的思想,基本上找到了与里奇在同一时期创建的张量乘法类似的定义。今天,它被称为“张量积”,或者按照格拉斯曼的说法,称为“外积”,尽管吉布斯和里奇没有使用过这些名字。它是一种将两个张量的信息合并为一个张量的方法——一种罗马数字的乘法版本,你可以添加更多符号来增加数字的大小:I、II、III、V、VI、VII、VIII 等等。这个类比还说明了外积通常不交换的事实:VI 与 IV 是不同的数字。

In the 1880s, Gibbs developed Grassmann’s idea further, essentially hitting on a similar definition of tensor multiplication as Ricci was creating around the same time. Today it is called a “tensor product” or, following Grassmann, an “outer product,” although Gibbs and Ricci didn’t use those names. It is a way of combining information from two tensors into one— a kind of multiplicative version of Roman numerals, where you add more symbols to increase the magnitude of a number: I, II, III, V, VI, VII, VIII, and so on. This analogy also illustrates the fact that outer products are generally not commutative: VI is a different number from IV.

您可以在图 11.1中看到张量(或外积)的概念——这也强调了向量和矩阵可以被视为张量的事实。这有一个复杂的数学原因,但现在只需注意到,就像张量一样,它们的分量也用索引表示——向量有一个索引,矩阵有两个索引,正如标题所示图 11.1说明了这一点。当然,正如我在第 9 章中暗示的那样,张量的标记不仅限于其分量用索引表示这一事实——但现在让我们先讨论这一点。因为张量积是一种产生新张量的方法,具有更多的索引,每个索引都会告诉你有关张量分量所表示的数据的一些具体信息。

You can see the idea of tensor (or outer) products in figure 11.1— which also highlights the fact that vectors and matrices can be thought of as tensors. There’s a sophisticated mathematical reason for this, but for now it’s enough to notice that like tensors, their components are represented with indices—one index for vectors, two for matrices, as the caption to figure 11.1 spells out. Of course, as I intimated in chapter 9, the mark of a tensor goes beyond the fact that its components are denoted with indices— but let’s go with this for now. For the tensor product is a way of producing new tensors, with even more indices, each index telling you something specific about the data represented by the tensor component.

图像

图 11.1张量(或外积)结合了被乘张量的信息。普通数字用符号表示,例如如果它们表示不依赖于坐标的量(例如温度),那么它们就是标量,而由于标量在坐标变化下不会改变,所以它们是张量。我们将看到向量也是张量,我们已经看到它们的分量用索引表示,以表示测量分量的轴。

FIGURE 11.1. Tensor (or outer) products combine information from the tensors being multiplied. Ordinary numbers are represented by a symbol such as a. If they represent quantities that don’t depend on coordinates—such as temperature—then they are scalars, and since scalars don’t change under coordinate changes, they are tensors. We’ll see that vectors are tensors, too, and we’ve already seen that their components are denoted with an index to represent the axis from which the component is measured.

列向量u和行向量v的张量积可以表示为一个矩阵。您可能熟悉表示矩阵第i行和第j列元素的符号a ij ,因此这里a ij = u i v j。我将在叙述中的 2-D 示例中详细说明这一点。以同样的方式,您可以构建更多的张量积。例如,1 索引向量u和矩阵A的张量积具有 3 个索引的分量,正如您在叙述中看到的那样。类似地,如果两个矩阵AB分别由向量外积形成,则它们的张量积的分量将是c ijklu i v j w k s l,依此类推。这不是构建新张量的唯一方法,但请注意,随着张量“秩”的增加,您可以表示更多信息。

The tensor product of a column vector u and a row vector v can be represented as a matrix. You’re likely familiar with the symbol aij for the element in the ith row and jth column of a matrix, so here aij = uivj. I’ll spell this out in the 2-D example in the narrative. In the same way, you can build up more tensor products. For example, the tensor product of the 1-index vector u and a matrix A has components with 3 indices, as you can also see in the narrative. Similarly, if two matrices A and B were each formed from the outer product of vectors, the components of their tensor product would be cijkluivjwksl, and so on. This is not the only way to build new tensors, but notice that you can represent more information as the tensor “rank” increases.

“秩”也称为张量的“阶”,因为里奇在提出这一概念时就是这么称呼的。它与从一个坐标系转换到另一个坐标系所需的变换矩阵的数量有关,但它本质上对应于索引的数量,每个索引代表不同类型的信息。(如果您熟悉矩阵代数,请注意,对于被视为张量的矩阵,“秩”的用法与线性代数中的用法不同。)我们将在后面进一步了解这一点。

The “rank” is also called the “order” of the tensor, for this is what Ricci called it when he introduced the idea. It has to do with the number of transformation matrices needed to transform from one coordinate system to another, but it essentially corresponds to the number of indices, each one representing a different type of information. (If you’re familiar with matrix algebra, note that for matrices viewed as tensors, this is a different use of “rank” from that in linear algebra.) We’ll see more about this as we go.

张量和数据科学(以及量子力学的概述)

TENSORS AND DATA SCIENCE (AND A PEEK AT QUANTUM MECHANICS)

图 11.1还说明了张量如何在当今的数据科学中存储和组合数据。在第 4 章中,我概述了向量和矩阵在机器学习和搜索引擎中的使用方式,但使用张量,你不仅可以添加更多数据,还可以添加不同类型的数据。例如,我们看到在搜索引擎中,信息可以存储为矩阵,行代表关键词,列代表包含这些关键词的不同文档。张量不仅允许你添加更多单词或更多文档(你可以通过增大矩阵来做到这一点),还允许你添加矩阵无法容纳的附加信息。例如,如果要添加文档的发布日期和作者,则必须将原始矩阵扩展为图 11.1所示的 4-D 形状。关键是每种类型的信息都有自己的索引。

Figure 11.1 also illustrates how tensors enable data to be stored and combined in data science today. In chapter 4, I outlined how vectors and matrices are used in machine learning and search engines, but with tensors you can add in not just more data but different kinds of data. For instance, we saw that in a search engine, information can be stored as a matrix, with rows representing key words, and columns representing different documents containing these key words. Tensors allow you not just to add more words or more documents—you can do that by making the matrix larger—but additional information that you can’t fit into the matrix. For example, if you want to add the date of publication of the document, and the author, you have to extend your original matrix to the 4-D shape shown in figure 11.1. The key thing is that each type of information has its own index.

张量的另一个现代应用是信号处理,例如用于解释脑电图 (EEG) 或心电图 (ECG)。除了信号的空间分量之外,您可能还想知道其时间和频率分量等信息——同样,每种类型的信息都需要自己的索引。

Another modern application of tensors is signal processing, which is used in interpreting an electroencephalogram (EEG) or electrocardiogram (ECG), for example. In addition to the spatial components of the signal, you might want to know such things as its temporal and frequency components—and again, each type of information requires its own index.

因此,分量的数量和索引的数量是有区别的。在普通的向量分析中,我们已经看到,在n维空间中(或者更准确地说,在n维“向量空间”中,即具有群属性的n维空间),一个向量有n 个分量,每个分量从n 个坐标轴之一测量。特定的轴为分量赋予特定的标签,因此一个向量有分量v 1v x,等等,直到v n。换句话说,你有n 个分量,但每个分量只有一个索引,并且这个索引采用n 个值之一,每个维度一个。我们还看到,矩阵的每个分量(或元素)都有两个索引:一个定位行,另一个定位列。你需要两个位置来确定特定元素的位置,因此你需要两个索引。类似地,我们在图 9.5中看到,应力张量的分量需要两个指标,一个指标表示应力作用的表面,另一个指标表示力;在三维空间中,每个索引取值从 1 到 3(或者从xz),因此共有 3 × 3 = 9 个分量,如图9.4所示。

So, there’s a difference between the number of components and the number of indices. In ordinary vector analysis we’ve seen that in n-dimensional space (or more properly, in an n-dimensional “vector space”— an n-D space with group properties), a vector has n components, each measured from one of the n coordinate axes. The particular axis gives the particular label on the component—so a vector has components v1vx, say, and so on up to vn. In other words, you have n components, but each has only one index—and this one index takes one of n values, one for each dimension. We’ve also seen that each component (or element) of a matrix has two indices: one locating the row and the other locating the column. You need both locations to pinpoint the position of a particular element, so you need two indices. Similarly, we saw in figure 9.5 that the component of a stress tensor needs two indices, one to indicate the surface on which the stress is acting, and the other to indicate the force; in three-dimensional space, each index takes values from 1 to 3 (or from x to z), so that’s 3 × 3 = 9 components, as we saw in figure 9.4.

高阶张量也是如此。例如,我在上一章中提到了黎曼张量,它有四个指标。乍一看,人们很容易认为它需要四个指标,因为它代表了 4 维时空的曲率——但我们在四维空间中工作,这一事实意味着黎曼张量上的每个指标可以取四个值,每个维度一个。指标本身代表黎曼张量的不同属性或构建块。9因此,虽然普通空间中的应力张量有 3 × 3 = 9 个分量,但时空中的 4 指标张量将有 4 × 4 × 4 × 4 = 256 个分量:四个指标中的每一个都有四个坐标选择。这是一个可以在一个张量中容纳的大量信息!并且您可以继续使用具有任意多维度和阶数的张量。 (但如果张量具有“对称性”,就像黎曼张量一样 - 例如交换或“反射”两个指标不会改变分量的值 - 它的某些分量是相同的,因此它可以表示的可能数据量就会减少。)

And so it goes for higher-order tensors. For example, I referred in the previous chapter to the Riemann tensor, which has four indices. It’s tempting at first glance to think that it needs four indices because it represents the curvature of 4-dimensional space-time—but the fact that we’re working in four dimensions just means that each index on the Riemann tensor can take four values, one for each dimension. The indices themselves each represent a different attribute or building block of the Riemann tensor.9 So, while the stress tensor in ordinary space has 3 × 3 = 9 components, a 4-index tensor in space-time will have 4 × 4 × 4 × 4 = 256 components: four coordinate choices for each of the four indices. That’s a lot of information you can fit into one tensor! And you can go on to tensors with as many dimensions and orders as you like. (But if a tensor has “symmetries,” as the Riemann tensor does—such as when interchanging or “reflecting” two indices doesn’t change the value of the component—some of its components are the same, so the possible amount of data it can represent is reduced.)

再举一个数字张量的应用,在图像处理中,每个像素的位置都表示在一个矩阵中,该矩阵的行和列代表图片的尺寸——但要生成彩色图像,需要三层矩阵,红、绿、蓝三色各一层。(这三种颜色是麦克斯韦发现彩色幻灯片摄影的遗产:他和他的助手托马斯·萨顿使用红、绿、蓝滤光片拍摄了有史以来第一张永久彩色照片,即我在第 6 章中提到的格子缎带。10因此,张量中编码相关像素信息的第三个索引代表颜色。同样,举另一个例子,如果将来自各种不同类型的测试的数据结合起来,医疗诊断会更准确,每个测试都由一个索引表示。这些多索引构造——这些张量及其乘积——在数据科学中非常重要,以至于谷歌将其一个机器学习平台命名为 TensorFlow,还有各种其他程序和工具,如 Tensorlab 和 Tensorly。

To take another digital tensor application, in image processing the location of each pixel is represented in a matrix whose rows and columns represent the dimensions of the picture—but to produce colour images three layers of matrices are needed, one for each of the colours red, green, and blue. (These three colours are a legacy of Maxwell’s discovery of colour slide photography: he and his assistant Thomas Sutton used red, green, and blue filters to take the first-ever permanent colour photograph, the tartan ribbon I mentioned in chap. 6.10) So, the third index in the tensor encoding the relevant pixel information represents the colour. Similarly, and taking a different example, medical diagnoses are more accurate if they combine data from a variety of different types of test, each represented by an index. These kinds of multi-index constructions—these tensors, and their products—are so important in data science that Google named one of its machine-learning platforms TensorFlow, and there are various other programs and tools, such as Tensorlab and Tensorly.

由于张量积非常重要,我将详细说明从列向量u和行向量v的张量积生成矩阵的想法,如图 11.1所示。为简单起见,我将向量设为二维:

Since tensor products are so important, I’ll spell out the idea of producing a matrix from the tensor product of a column vector u and a row vector v, as in figure 11.1. For simplicity, I’ll make the vectors 2-D here:

1212=11122122

u1u2v1v2=u1v1u1v2u2v1u2v2.

这是将两个向量的信息组合起来的巧妙方法,并且在这种情况下也遵循普通矩阵乘法的规则。但是当你尝试将u乘以 2 × 2 矩阵时,你可以看到矩阵乘法和张量积之间的区别:普通矩阵规则根本不允许你将 2 × 1 矩阵乘以 2 × 2 矩阵,但张量积却可以:

This is a neat way of combining the information from two vectors, and it also follows the rules of ordinary matrix multiplication in this case. But you can see the difference between matrix multiplication and tensor products when you try to multiply u by a 2 × 2 matrix: the ordinary matrix rules don’t allow you to multiply a 2 × 1 matrix by a 2 × 2 one at all, but the tensor product does:

12一个11一个12一个21一个22=1一个11一个12一个21一个222一个11一个12一个21一个22=1一个111一个121一个211一个222一个112一个122一个212一个22

u1u2a11a12a21a22=u1a11a12a21a22u2a11a12a21a22=u1a11u1a12u1a21u1a22u2a11u2a12u2a21u2a22.

因此,张量积是一种将来自两个(或更多)系统或集合的信息组合成一个大系统或集合的方法。例如,考虑一下自然语言处理 (NLP) 程序,它们支持许多奇迹,例如电子邮件垃圾邮件过滤器、语言翻译器、将口语单词转换为文本(供听力障碍者使用,例如为电视节目加字幕)、GPS 上的有用语音、公司聊天机器人的礼貌文本、网络搜索中使用的预测文本,以及电子邮件和 Instagram 等应用程序,以及 OpenAI 的 ChatGPT 等机器人生成的非常像人类的文本。张量积可以提供一种将一组单词与一组语法指令相结合的方法——其中单词通过在单词词典中分配位置来表示为向量,语法也是如此。我应该在这里补充一点,用于开发复杂 NLP(特别是大型语言模型或 LLM)的大部分训练数据都是在未经内容创建者知情或许可的情况下从网络上抓取的人工生成内容,我很高兴看到作家和艺术家们正在尝试反击这种现象盗窃。11由于张量本身不是问题,并且由于 NLP 和 LLM 程序和应用程序既有好处也有问题我将继续尝试这种张量积的出色应用。

So, the tensor product is a way of combining information from two (or more) systems or sets into a single large one. For instance, consider the natural language processing (NLP) programs behind such marvels as email spam filters, language translators, converting spoken words to text (for use by those with hearing problems, for example, such as in captioning TV programs), the helpful voice on your GPS, the polite text from a company’s chatbot, the predictive text used in web searches, say, and apps such as email and Instagram, and the spectacularly human-like text generated by bots such as OpenAI’s ChatGPT. Tensor products can offer a way to combine a set of words with a set of grammatical instructions—where words are represented as vectors by assigning them a position within a dictionary of words, and similarly for the grammar. I should add here that much of the training data used to develop sophisticated NLP (especially large language models or LLMs) is human-generated content scraped from the web without the content creators’ knowledge or permission, and I’m heartened that writers and artists are attempting to fight back against this theft.11 But since tensors themselves are not the problem, and since there are benefits as well as problems12 with NLP and LLM programs and applications, I’ll venture on with this brilliant application of tensor products.

简单来说,假设词典包含三个单词“cats、love、mice”,每个单词都分配有一个从 1 到 3 的位置编号。因此,这些单词的向量表示的维度为三,所以如果“cats”在第一个位置,“love”在第二个位置,“mice”在第三个位置,那么这些词将表示为向量 (1,0,0)、(0,1,0)、(0,0,1),我将其分别标记为C、L、M。要创建句子“Cats love mice”,只需添加向量:C + L + M = (1,1,1)。但这与“Mice love cats”的向量表示没有什么不同,事实绝对不是这样。所以,张量积可以派上用场了。取第二组向量,表示这些单词将扮演的角色的关键语法指令:主语、宾语、动词,标记为S、O、V,分别表示为 (1,0,0)、(0,1,0) 和 (0,0,1)。“猫”(表示为列向量)和“主语”(行向量)的张量积将表示为我们之前看到的列向量和行向量:

To keep it simple, suppose the word dictionary contains the three words, “cats, love, mice,” and each word is assigned a position number from 1 to 3. The dimension of the vector representations of these words will therefore be three, so if “cats” is in the first position, “love” is in the second, and “mice” is in the third, then these words would be represented as the vectors (1,0,0), (0,1,0), (0,0,1), which I’ll label C, L, M, respectively. To create the sentence “Cats love mice,” just add the vectors: C + L + M = (1,1,1). But this is no different from the vector representation of “Mice love cats,” which is definitely not the case. So it’s here that tensor products can come to the rescue. Take a second set of vectors, representing key grammatical instructions for the roles these words will play: subject, object, verb, labeled as, say, S, O, V, and represented as (1,0,0), (0,1,0), and (0,0,1), respectively. The tensor product of “cats” (represented as a column vector) and “subject” (a row vector) would be represented as we saw just before for column and row vectors:

100100=100000000

100100=100000000,

对于“mice”和“object”以及“like”和“verb”的张量积,也是类似的。现在你可以构造一个明确的句子,

and similarly for the tensor products of “mice” and “object,” and “like” and “verb.” Now you can construct an unambiguous sentence,

年代+大号+=100001010

CS+LV+MO=100001010,

其中 ⊗ 是张量积的符号。这与“老鼠爱猫”不同:

where ⊗ is the symbol for tensor products. This is different from “Mice love cats”:

年代+大号+=010001100

MS+LV+CO=010001100.

这句话虽然是假的,但却不含糊

Although this sentence is false, it is unambiguous.

这个例子只是在 NLP 中应用张量积的一种方法。13量子力学中应用它们的一种方法是表示几个粒子的“量子态”。

This example is just one way of applying tensor products in NLP.13 And one way of applying them in quantum mechanics is in representing the “quantum state” of several particles.

我们在序言中看到,电子的自旋可以是“向上”的,用矢量 (1, 0) 表示,也可以是“向下”的,用矢量 (0, 1) 表示;因此,这两种可能性的“叠加”——一种可能出现任何结果的中间状态——是 (α, β),其中 α 是处于“向上”状态的“权重”或概率幅,β 是处于“向下”状态的概率幅。(“概率幅”只是意味着 |α 2 | + |β 2 | = 1。为了使概率成立,α 和 β 是复数。)我们还看到,自旋可用于表示量子计算中的 0 和 1,自旋“向上”状态表示二进制数字 0,而“向下”表示 1。为了方便起见,我将这些“向上”和“向下”状态写为行向量,但在量子力学中,状态向量写为列向量,它们通常被称为“kets”。在第 4 章中,我谈到了单位向量i、j、k是构造向量v的“基础” ——类似地,电子或量子比特的自旋态 ψ 的基础可以选择为10表示自旋“向上”状态,01自旋“向下”。使用以量子先驱保罗·狄拉克命名的“狄拉克符号”,这些基向量用 kets |0〉和 |1〉表示。因此,量子比特的状态向量以分量形式表示为

We saw in the prologue that the spin of an electron can be “up,” represented by the vector (1, 0), or “down,” represented by (0, 1); so a “superposition” of these two possibilities—an in-between state in which either outcome is possible—is (α, β), where α is the “weight” or probability amplitude of being in the “up” state and β the probability amplitude of being in the “down” state. (“Probability amplitude” just means that |α2| + |β2| = 1. To make the probabilities work, α and β are complex numbers.) We also saw that spin can be used to represent the 0s and 1s in quantum computing, with the spin “up” state representing the binary digit 0, say, and “down” representing 1. I wrote these “up” and “down” states as row vectors for convenience, but in quantum mechanics state vectors are written as column vectors, and they are often called “kets.” In chapter 4 I spoke about the unit vectors i, j, k being a “basis” for constructing a vector v—and analogously, the basis for the spin state ψ of an electron, or a qubit, can be chosen so that 10 represents the spin “up” state, and 01 for spin “down.” Using what is known as “Dirac notation,” after quantum pioneer Paul Dirac, these basis vectors are denoted by the kets |0〉 and |1〉. So the state vector for a qubit is represented in component form as

|ψα|0+β|1

|ψα|0+β|1.

这意味着,在实际观察到之前,量子比特处于 |0〉 和 |1〉 两种状态的叠加状态,α 和 β 分别表示处于每种状态的概率。当量子比特组合时,张量积就会发挥作用,当然,要制造出可用的量子计算机,就必须组合它们。不过,为了更轻松一点,我们只取两个量子比特,并将它们的状态表示为

What this means is that until it is actually observed, the qubit is in a state of superposition between the two states |0〉 and |1〉, α and β representing the respective likelihoods of it being in each state. Tensor products come into it when qubits are combined, as of course they must be to make a usable quantum computer. To go gently, though, let’s take just two qubits, and represent their states as

|ψ1αβ,|ψ2=γδ

|ψ1αβ,|ψ2=γδ.

为了找到这两个系统的组合状态,取它们的张量积:

To find the state of the combination of these two systems, take their tensor product:

|ψ=|ψ1|ψ2=αγαδβλβδ

|ψ=|ψ1|ψ2=αγαδβλβδ.

它是一个 4 × 1 向量,给出四种可能性的概率幅度:两个量子比特都处于向上状态(都表示零)、第一个向上而另一个向下、第二个向上而第一个向下以及两个都处于向下状态。每个量子比特都表示 0 和 1 的叠加,这种能力使得量子计算机具有如此强大的潜力,因为它们可以同时执行多个计算。您可以从以下事实中感受到这种强大能力:在n 个量子比特的系统中,张量积将具有 2 n 个复数分量。即使量子比特数量相对较少,也会同时处理大量 0 和 1。14

It’s a 4 × 1 vector giving the probability amplitudes for four possibilities: both qubits are in the up state (both represent zeroes), the first is up and the other is down, the second is up and the first is down, and both are in the down state. It’s this ability for each qubit to represent a superposition of 0s and 1s that makes quantum computers so potentially powerful, for they can carry out multiple computations at the same time. You can sense this power from the fact that in a system of n qubits, the tensor product will have 2n complex-number components. Even with a relatively modest number of qubits that’s a lot of 0’s and 1’s being processed simultaneously.14

• • •

• • •

表示量子态的列向量称为“kets”,表示为任意状态的 | A〉,而行向量称为“bras”,表示为〈A |。这是因为当你把 bra 和 ket 放在一起时——例如当你取两个状态 | A〉和 | B 〉的标量(或内积)时——你就完成了括号(“bra-ket”):

While column vectors representing quantum states are called “kets,” represented as |A〉 for an arbitrary state, row vectors are called “bras,” denoted by 〈A|. That’s because when you put a bra and a ket together—such as when you take the scalar (or inner) product of two states |A〉 and |B〉 —you complete the bracket (“bra-ket”):

|一个

B|A.

标量积是状态向量“归一化”所必需的,以确保 α 和 β 等权重确实与测量结果的概率相关——但这里的重点是B〉 列向量的内容现在充当 bra 或行向量 〈B |,对 ket | A〉 进行运算以得出标量积。这听起来很复杂(从技术上讲,bra 是 ket 的“对偶”,两者都位于复向量空间中)——但您可以将其视为普通向量分析结果的应用,其中向量可以发挥不同的作用,具体取决于您如何编写它们。

Scalar products are needed in the “normalisation” of state vectors that ensures weights such as α and β do relate to the probabilities of a measurement result—but the point here is that the content of the B〉 column vector is now acting as a bra or row vector, 〈B|, operating on the ket |A〉 to give the scalar product. This sounds complicated (and technically, a bra is a “dual” of a ket, and both reside in complex vector spaces)—but you can look at it as an application of a result from ordinary vector analysis, where vectors can play different roles, depending on how you write them.

例如,回到我们的列向量u和行向量v,我们之前看到,u乘以v的普通矩阵乘法给出一个 2 × 2 矩阵(在这种情况下也是它们的张量积)。但是如果你交换顺序,v乘以u的普通矩阵乘法给出的不是一个矩阵而是一个数字,v 1 u 1 + v 2 u 2。事实上,它是两个向量的标量积。(至少,它是平坦二维欧几里得空间中的标量积。正如我在第 9 章中提到的,标量积取决于度量。例如,在闵可夫斯基时空中,标量积是

For instance, going back to our column vector u and row vector v, we saw a bit earlier that ordinary matrix multiplication of u times v gives a 2 × 2 matrix (which in this case is also their tensor product). But if you swap the order, ordinary matrix multiplication of v by u gives not a matrix but a number, v1u1 + v2u2. In fact, it is the scalar product of the two vectors. (At least, it is the scalar product in flat two-dimensional Euclidean space. As I mentioned in chapter 9, the scalar product depends on the metric. In Minkowski space-time, for example, the scalar product is

a b = a 1 b 1 + a 2 b 2 + a 3 b 3 a 4 b 4,或a b = − a 0 b 0 + a 1 b 1 + a 2 b 2 + a 3 b 3

ab = a1b1 + a2b2 + a3b3a4b4, or ab = − a0 b0 + a1b1 + a2b2 + a3b3

如果时间分量用 0 下标而不是 4 表示,则为 0。因此,您可以看到,将向量(数据)写为行还是列会产生影响。这种事情可能会让数学看起来怪异且自相矛盾——但正是这种细节激起了富有创造力的数学家的好奇心。里奇和他的继任者想出了一个巧妙的解决方法。

if the time component is denoted by a 0 subscript rather than a 4). So, you can see that it makes a difference whether you write your vector, your data, as a row or column. This is the kind of thing that might make maths seem bizarre and contradictory—but it’s just this sort of detail that piques a creative mathematician’s curiosity. And Ricci and his successors came up with a neat way around it.

首先,用不同的符号表示这两种向量。按照里奇的做法,用“楼上指标”或上标来表示列向量的分量——因此,你不再写u 1u 2,而是u 1u 2。(实际上,他将楼上的指标放在括号中,大概是为了清楚地表明这些是标签,而不是幂。随着爱因斯坦和格罗斯曼等数学家和物理学家越来越擅长使用指标符号,他们放弃了括号。)保留行向量的楼下指标,即下标。因此,对于由列向量乘以行向量形成的矩阵,元素可以表示为u 1 v 1u 1 v 2u 2 v 1u 2 v 2等等。只需看一下符号,您就能立即看出您正在将两种不同类型的矢量相乘。几十年后,狄拉克通过他的 bra 和 ket 符号应用了这种区别。

First, represent the two types of vector with different notation. Following Ricci, write the components of column vectors with an “upstairs index” or superscript—so you no longer write u1 and u2 but u1 and u2. (Actually he put upstairs indices in brackets, presumably to make it clear that these are labels, not powers. As mathematicians and physicists such as Einstein and Grossmann became more adept at using index notation, they discarded the brackets.) Keep the downstairs indices, the subscripts, for the row vectors. So, in the case of a matrix formed from a column vector times a row vector, the elements can be represented as u1v1, u1v2, u2v1, u2v2, and so on. Straightaway you can see, just by looking at the notation, that you are multiplying two different kinds of vector. Decades later, Dirac would apply this distinction via his bra and ket notation.

其次,给这两个实体赋予不同的名称,以避免混淆。今天,在这种情况下,“向量”一词指的是列向量,而行向量被称为“单形式”或“对偶向量”。二十世纪早期的研究人员创造了这些术语;这个想法源于格拉斯曼,他使用了“补体”而不是“对偶”一词。胸罩是单形式的例子。早在 19 世纪 80 年代,Ricci 就将向量和单形式分别称为“逆变向量”和“协变向量”,这些名称也沿用至今。

Second, give these two entities different names, to avoid confusion. Today the word “vector” in this context refers to column vectors, while row vectors are called “one-forms” or “dual vectors.” Early twentiethcentury researchers coined these terms; the idea originates with Grassmann, who had used the term “complement” instead of “dual.” Bras are examples of one-forms. Back in the 1880s, Ricci called vectors and one-forms “contravariant vectors” and “covariant vectors,” respectively, and these names are also used today.

实际上,Ricci 并没有专门讨论行向量和列向量,因为这些只是他提出的两种张量的示例。正如我们将在接下来的两节中看到的那样,这种区别背后的一般思想(以及 Ricci 选择的名称背后的思想)来自超越指标符号和张量积的概念,而这通常是数据科学中所需的张量数学的主要方面。

Actually, Ricci didn’t talk specifically about row vectors and column vectors, for these are just examples of his two types of tensor. As we’ll see in the next two sections, the general idea behind this distinction—and behind Ricci’s choice of names—comes from a concept that goes beyond index notation and tensor products, which are often the main aspects of tensor maths needed in data science.

事实上,在数据科学中,索引的位置并不像在数学和物理学中那么重要,因为通常输入数据时根本不需要任何抽象符号。相反,许多程序员使用秩和“形状”来表征每个不同的张量。例如,在 TensorFlow(以及其他编程语言库,如 Python 的 Numpy)中,形状指的是维度。标量的形状为 0,表示为空括号:[ ]。向量被编程为一串数字,每个组件一个数字;它的形状是组件的数量,因此 3 维向量的形状为 [3]。秩 2 张量可以表示为矩阵,它们的形状是行数和列数,因此上面的 2 × 2 矩阵的形状为 [2, 2]。秩 3 张量(例如 2 × 3 × 5 数组)的形状为 [2, 3, 5],依此类推。

In fact, even the position of indices is not so important in data science as it is in maths and physics, for often the data are entered without any need for abstract symbols at all. Rather, it’s the rank and “shape” that many programmers use to characterise each different tensor. For example, in TensorFlow (and other programming language libraries such as Python’s Numpy), the shape refers to the dimension. A scalar has a shape 0, represented as an empty bracket: [ ]. A vector is programmed as a string of numbers, one for each component; its shape is the number of components, so a 3-D vector has shape [3]. Rank 2 tensors can be represented as matrices, and their shape is the number of rows and columns, so the 2 × 2 matrix above would have shape [2, 2]. A rank 3 tensor, such as a 2 × 3 × 5 array, has shape [2, 3, 5], and so on.

强调形状的一个优点是,程序员可以包含 TensorFlow 所称的“不规则张量”,即包含不同大小字符串的数组——可能是一串单词或句子,其中字母或单词的数量与空间维度无关。正如我所提到的,在n维空间中,张量上的每个索引必须取 1 到n之间的值,但“不规则张量”可以具有可变的大小。这是数据科学家根据自己的需要调整数学概念的一个很好的例子。

An advantage of emphasising shape is that programmers can include what TensorFlow calls “ragged tensors,” arrays with strings of different sizes—a string of words or sentences, perhaps, where the number of letters or words has nothing to do with the dimension of space. As I’ve mentioned, in n-dimensional space, each index on a tensor must take a value between 1 and n, but “ragged tensors” are allowed to have variable sizes. This is a nice example of the way data scientists have adapted a mathematical concept for their own needs.

这一切与张量的最初使用方式相去甚远,尽管是无意的——作为数学物理中的应力和度量张量。不过,这些早期的用途也与表示和处理信息有关。我说的“处理”是指知道如何将信息组合成新的张量,以及如何解释和应用结果。因此,要想获得张量的称号,张量,仅仅将信息放入列表或数组是不够的——四千年前美索不达米亚人就做这种事了。要成为张量,数组必须遵循某些规则,就像我们在第 4 章中看到的向量和矩阵一样。我们已经见过张量积,当然还有加法的规则。我们看到了向量加法的平行四边形规则如何考虑它们的大小和方向,但更一般地说,向量和张量的加法——以及乘法——必须遵循“线性”定律。(这意味着,例如

All this is a long way from the way tensors were first used, albeit unwittingly—as stress and metric tensors in mathematical physics. Still, these earlier uses also had to do with representing and handling information. By “handling” I mean knowing how to combine information into new tensors and how to interpret and apply the results. So, to earn the title of tensor, it’s not enough simply to put information into a list or array—the Mesopotamians were doing that sort of thing four thousand years ago. To be a tensor, the arrays have to obey certain rules, just as we saw with vectors and matrices in chapter 4. We’ve already met tensor products, and there are also rules for addition, of course. We saw how the parallelogram rule for adding vectors takes account of their magnitude and direction, but more generally, vector and tensor addition—and multiplication, too— must obey the laws of “linearity.” (This means, for example, that

(2 a)∙ b = 2(ab)= a ∙(2 b),且a ∙(u + v)= au + av

(2a) ∙ b = 2(ab) = a ∙ (2b), and a ∙ (u + v) = au + av

—类似于算术中的分配律。)

—analogous to the distributive law in arithmetic.)

但数学和物理学中最重要的是张量能够以不变的方式(绝对不变地)表示信息,而不会因坐标选择而产生虚假数据。对于许多数据科学应用来说,这不是问题,尽管在某些应用中,比如我们之前看到的神经网络,这很重要。无论如何,张量的概念远比图 11.1所显示的要多得多。

But the most important thing in maths and physics is the ability of tensors to represent information invariantly—“absolutely”—without spurious data coming from the choice of coordinates. This is not an issue for many data science applications, although it is important in some of them, such as the neural networks we saw earlier. Either way, there’s much more to the idea of a tensor than figure 11.1 suggests.

不变性(以及学术丑闻)

INVARIANCE (AND AN ACADEMIC SCANDAL)

不变量的概念(即在旋转、平移或以其他方式改变参考系时保持不变的事物)至少自 19 世纪 40 年代凯莱和布尔合作以来就一直吸引着数学家。在第 9 章中,我们看到了许多不变性的例子,从雪花的形状到标量积ab再到闵可夫斯基度量,

The idea of invariants—things that stay the same when you rotate, translate or otherwise change your frame of reference—had been intriguing mathematicians at least since Cayley and Boole were working together in the 1840s. In chapter 9 we saw many examples of invariance, from the shape of snowflakes to the scalar product ab to the Minkowski metric,

ds 2 = dx 2 + dy 2 + dz 2 − ( cdt ) 2

ds2 = dx2 + dy2 + dz2 − (cdt)2

(其中单位通常选择为c = 1)。但是,正如我们在图 9.19.3中看到的那样,这些形状和表达式中的每一个仅对特定的一组坐标变换不变。

(where units are often chosen so that c = 1). But each of these shapes and expressions is invariant only with respect to a particular group of coordinate transformations, as we saw in figures 9.1 and 9.3.

在研究不变量时,黎曼等数学家着眼于物理应用——在他看来,是曲面的曲率——而其他人,包括凯莱和克莱因,则对这些坐标变换群的纯数学结构感兴趣。克莱因曾担任《数学年鉴》杂志的主编,职位他的导师阿尔弗雷德·克莱布施 (Alfred Clebsch) 突然死于白喉,他便开始从事这一研究。1869 年,当尤斯图斯·格拉斯曼 (Justus Grassmann) 以学生身份来到哥廷根时,克莱布施教授曾将他父亲的《Ausdehnungslehre》的副本交给他。克莱布施教授对此印象深刻,他继续拓展格拉斯曼的思想,并共同创办了《数学年鉴》,作为研究不变理论的渠道。

In studying invariants, mathematicians such as Riemann had their eye on physical applications—in his case, the curvature of surfaces—while others, including Cayley and Klein, were interested in the purely mathematical structure of these groups of coordinate transformations. Klein had been the managing editor of the journal Mathematische Annalen, a position he took up when his mentor Alfred Clebsch suddenly died of diphtheria. Clebsch is the professor to whom Justus Grassmann had given a copy of his father’s Ausdehnungslehre when he arrived as a student at Göttingen in 1869. Clebsch was impressed, and he went on both to extend Grassman’s ideas and to cofound the Mathematische Annalen as an outlet for research on invariant theory.

我可以将许多其他名字添加到 20 世纪下半叶致力于不变量和/或微分形式的数学家名单中,包括里奇在比萨的前数学教授恩里科·贝蒂。这里有一个重叠的线索,因为贝蒂的成就之一是将斯托克斯定理推广到n维。我们之前看到,这个定理的原始 3-D 版本首次发表在 1854 年的史密斯奖考试中,麦克斯韦参加了考试,并且它将曲面积分与线积分联系起来。关键是它以不变的方式实现这一点 - 无论使用什么坐标,都应该得到相同的表面积。因此,贝蒂在这方面的工作是数学家对坐标变换和不变性感兴趣的多种原因的一个例子。

There are many other names I could add to the list of mathematicians working on invariants and/or differential forms during the latter half of the century—including Ricci’s former maths professor at Pisa, Enrico Betti. There’s an overlapping thread here, for one of Betti’s achievements was his generalisation of Stokes’s theorem to n dimensions. We saw earlier that the original 3-D version of this theorem had been first published in the 1854 Smith’s Prize exam, which Maxwell sat—and that it relates a surface integral to a line integral. The key thing is that it does this in an invariant way— you should get the same surface area no matter the coordinates you use. So Betti’s work on this is an example of the diverse reasons mathematicians were interested in coordinate transformations and invariance.

贝蒂还让我们想起了 19 世纪欧洲发生的革命动乱。1848 年,作为一名学生,他参加了意大利独立战争的两次战役——他的论文导师曾领导托斯卡纳大学营。该营败给了奥地利人,但幸运的是,贝蒂幸存了下来,并成为一名重要的数学家和教师——正是在他的建议下,里奇去了柏林跟随克莱因学习。15贝蒂还为意大利新出版的纯数学和应用数学杂志《Annali di Matematica pura e applicata》撰稿。《Annali》和《Clebsch》以及克莱因的《Annalen》等专业期刊是数学家发表其工作成果的重要场所——如果这些期刊在托马斯·哈里奥特的时代就存在,也许他的工作成果就不会丢失这么久:世界上最早的现代科学期刊之一是伦敦皇家学会的《哲学学报》,它创刊于十七世纪六十年代,距离哈里奥特去世已过去了近半个世纪。

Betti also reminds us of the revolutionary upheavals taking place in Europe in the nineteenth century. As a student in 1848 he’d fought in two of the first battles for Italian independence—his thesis advisor had led the Tuscany university battalion. It lost, to the Austrians, but luckily Betti survived, and went on to become a significant mathematician and teacher— and it was on his advice that Ricci had gone to Berlin to study with Klein.15 Betti also contributed to the new Italian journal for pure and applied maths, Annali di Matematica pura e applicata. Specialist journals, such as Annali and Clebsch and Klein’s Annalen, were important places for mathematicians to publish their work—and had they existed in Thomas Harriot’s day, perhaps his work would not have been lost for so long: one of the world’s first modern scientific journals was the Philosophical Transactions of the Royal Society of London, which began in the 1660s, nearly half a century after Harriot’s death.

拥有成熟的大学、科学协会和期刊的国家往往是数学进步的中心,1884 年,里奇在《年鉴》上发表了他关于不变性的一些早期成果。他刚开始这段旅程时并不知道黎曼的工作。相反,他最初的灵感来自埃尔温·克里斯托费尔 (Elwin Christoffel) 1869 年论文的纯数学方法。如前所述,克里斯托费尔和黎曼都独立发现了现在所谓的克里斯托费尔符号和黎曼 (或黎曼-克里斯托费尔) 张量,正如黎曼在 1861 年的论文中所展示的,它给出了判断曲面是否平坦的不变条件。克里斯托费尔根本没有提到曲率——除了在论文末尾的注释中说,在 1854 年的任教资格论文中,黎曼将二次微分形式应用于线元素。但这对里奇来说已经足够了,他开始寻找黎曼的论文。

Countries with established universities, scientific societies, and journals tended to be at the centre of mathematical progress, and in 1884 Ricci published some of his first results on invariance in the Annali. He didn’t know of Riemann’s work when he first set out on this journey. Instead, he took his initial inspiration from the purely mathematical approach of Elwin Christoffel’s 1869 paper. As I’ve mentioned, Christoffel and Riemann both independently discovered what are now called the Christoffel symbols and the Riemann (or Riemann-Christoffel) tensor, which, as Riemann showed in his 1861 paper, gives the invariant condition for knowing whether a surface is flat or not. Christoffel didn’t refer to curvature at all—other than in a note at the end of his paper, saying that in his 1854 habilitation thesis Riemann had applied quadratic differential forms to the line element. But that was enough for Ricci, who went searching for Riemann’s papers.

与此同时,他还要教书——对于像利玛窦这样内敛、腼腆的人来说,这并非易事。但他对自己的学科充满热情,尽管他的讲座缺乏色彩,但仍然清晰而严谨——利玛窦是证明数学的佼佼者。正如他告诉一位同事的那样,“我不否认,当向那些不幸不认真对待所学知识的学生展示我的讲座中的证明时,它们会带来一些困难。”但他继续说,这并不能阻止他以应有的方式展示数学。毕竟,“如果我自己的判断没有欺骗我,这些证明是美丽的。”此外,最好的学生受到这种严谨的启发,他认为,其余的学生可能会从中受益,因为他们来自中学,数学基础不够扎实。16相信今天的许多讲师都能理解他。

At the same time, he had his teaching—not always an easy task for a reserved, diffident person like Ricci. But he was passionate about his subjects, and if his lectures lacked colour, they were nonetheless clear and rigorous—Ricci was a great one for proof. As he told a colleague, “I don’t deny that the proofs [in my lectures cause] some difficulty when they are presented to students who unfortunately don’t take their education seriously.” Still, he went on, that wasn’t going to stop him from presenting maths the way it should be presented. After all, “if my own judgment doesn’t deceive me, these proofs are beautiful.” Besides, the best students were inspired by such rigour, and the rest, he thought, might benefit from it, for they had come from secondary school with insufficient grounding in mathematical fundamentals.16 I’m sure many lecturers today can empathise with him.

利玛窦还忙于申请晋升为正教授。1884 年,利玛窦第一次申请时,他和年轻的竞争对手朱塞佩·维罗内塞都输给了一位更资深的候选人。利玛窦很乐意按照资历排序,但维罗内塞提出上诉。他还指出,他来自工人阶级(不像利玛窦),需要增加薪水来“帮助我可怜的父母和兄弟们”。当局善意对待他的请求,并给他提供了一个永久的教职,还增加了薪水。利玛窦没有慌乱,1887 年再次申请正教授时,他满怀希望。当时他已经 34 岁,在学术界享有盛誉。利玛窦的出版记录非常出色,但维罗内塞也申请了同样的职位。随后的争斗成为头条新闻,有传言称教职人员的欺诈行为最终剥夺了利玛窦应得的晋升。这场闹剧持续了三年,利玛窦终于获得了教授职位。17

Ricci was also busy applying for promotion to a full professorship. On his first attempt, in 1884, he and his young rival, Guiseppe Veronese, missed out to a more senior candidate. Ricci was happy to defer to seniority, but Veronese appealed. He also pointed out that he was from a working-class background (unlike Ricci) and needed the increased salary to “help my poor parents and my brothers.” The plea was treated kindly by the authorities, and he was given a permanent teaching position with an increase in pay. Unruffled, Ricci was hopeful when he tried again for a full professorship in 1887. He was now thirty-four and had a reputable publishing record—but so did Veronese, who applied for the same position. The ensuing battle made front-page news, with rumours that faculty skullduggery ultimately deprived Ricci of his rightful promotion. The saga dragged on, and it would take another three years for Ricci finally to gain his professorship.17

与此同时,作为一名杰出的数学家,他除了全身心投入到最终能使他的名字永垂不朽的工作中之外,还能做什么呢?

Meantime, what else could he do, brilliant mathematician that he was, but throw himself into the work that would, ultimately, immortalise his name?

如何处理所有这些指数

HOW TO HANDLE ALL THOSE INDICES

在接下来的几页中,我将花时间展示为什么里奇有两种不同的向量——或者用现代的术语来说,一个向量和一个形式——因为它们是所有张量的原型。 (实际上,他从一般的张量开始,并以向量为例:可能是因为当时的向量分析还没有今天这么完善,也是因为他的理论主要源于不变微分形式的研究,而不是向量分析。)这两种类型的向量与里奇的指标符号有关,这是一个绝佳的例子,说明了数学家如何使用符号来揭示数学概念的底层结构。我们在第 1 章中从代数的兴起和第 8 章中关于向量符号之争中看到了这一点。不过,这里的细节和方法可能对你来说不熟悉,尽管你需要的数学工具只是向量和矩阵乘法。

Over the next few pages, I’m going to spend time showing why Ricci had two different kinds of vector—or in modern terms, a vector and a oneform—because they are prototypes for all tensors. (Actually, he started with general tensors and gave vectors as an example: perhaps because vector analysis wasn’t so well established then as it is today, but also because his theory arose primarily from the study of invariant differential forms rather than from vector analysis.) These two types of vector relate to Ricci’s index notation, which is a marvelous example of the way mathematicians use symbolism to bring out the underlying structure of mathematical concepts. We saw something of this with the rise of algebra in chapter 1 and the war over vector notation in chapter 8. Here, though, there’s detail and an approach that may be unfamiliar to you, although the only maths tools you need are vector and matrix multiplication.

不过,和往常一样,如果你已经到了只想相信我的话的地步,那就跳到下一节。如果你到了下一节,决定只想看看爱因斯坦和格罗斯曼是如何用张量来制定广义相对论框架的,那就继续读下一章吧。里奇会理解的:学习一项新技能总是需要付出努力,他在“绝对微分学”的开创性概述的导言中写道。18泰特在试图让人们相信四元数的优势时也说过类似的话。但和泰特一样,里奇确信,在“克服了入门的困难”之后,读者很快就会相信这些方法的“优雅和清晰”——这些方法取决于对思想的正确表示不变性。这是一种依赖于发现模式的符号,因此它也相当有趣。

As always, though, if you get to the point where you just want to take my word for it, skip down to the next section. And if, once you get there, you decide you just want to see how Einstein and Grossmann used tensors in formulating the framework for general relativity, then go ahead to the next chapter. Ricci would understand: it always takes effort to learn a new skill, he wrote in the introduction to his seminal overview of his “absolute differential calculus.”18 Tait had said something similar when he was trying to convince people of the advantages of quaternions. But like Tait, Ricci was sure that after “surmounting the difficulties of initiation,” readers would soon convince themselves of the “elegance and clarity” of these methods—methods that hinge on the right notation for the idea of invariance. It’s a notation that relies on spotting patterns, so it’s also rather fun.

• • •

• • •

如果张量要体现坐标系变化下的不变性,里奇就必须找到一个张量在某一框架中的分量与同一张量在新框架中的分量之间的具体关系。我们在图 9.1中看到了位置向量的一个隐含示例,其中我从几何上证明了a的大小以及标量积ab在旋转坐标轴时不变。我当时没有展示每个向量分量如何变化,但这正是里奇试图回答的问题。

If tensors were to encode the idea of invariance under coordinate changes, Ricci had to find the specific relationship between the components of a tensor in one frame and those of the same tensor in the new frame. We saw an implicit example of this for the position vectors in figure 9.1, where I showed geometrically that the magnitude of a, and also the scalar product ab, are invariant when you rotate the coordinate axes. What I didn’t show then was how each vector component changed, but this is the kind of question that Ricci was trying to answer.

他首先概括了坐标变换的工作方式。但我将从一个具体的例子开始,即图 9.1中的旋转,其中从通常的x - y坐标到旋转的x′ - y′坐标的变换方程为:

He began by generalising the way coordinate transformations work. But I’ll start with a specific example, the rotation in figure 9.1, where the transformation equations from the usual x-y coordinates to the rotated x′-y′ ones are:

x′ = x cos θ + y sin θ,y′ = − x sin θ + y cos θ。

x′ = x cos θ + y sin θ, y′ = −x sin θ + y cos θ.

您可能已经注意到,这些是线性方程(它们仅包含xy,没有幂或其他乘积),并且您可以将它们写成矩阵方程:

You might have noticed already that these are linear equations (they contain just x and y, with no powers or other products), and that you can write them as a matrix equation:

=余弦θθθ余弦θ

xy=cosθsinθsinθcosθxy.

这是数学家在表示和处理信息时一次又一次使用向量和矩阵的例子。(您可能还注意到,这里的旋转矩阵与图 4.2中的旋转矩阵类似,但那里我们旋转的是机械臂,而这里,如图9.1所示,向量保持不变,但轴本身会旋转。)

This is another instance of the way vectors and matrices pop up again and again when mathematicians want to represent and handle information. (You might also have noticed that the rotation matrix here is similar to the one in fig. 4.2, but there we were rotating the robot arm, whereas here, as in fig. 9.1, the vector stays the same, but the axes themselves rotate.)

如果你让=A表示“变换矩阵”,可以更经济地将此坐标变换方程写为:

If you let X=xy and let A represent the “transformation matrix,” you can write this coordinate transformation equation more economically as:

X ′= AX

X′ = AX.

正如我们在第 1 章中看到的,当代数学家开始使用符号而不是单词或具体的数值示例时,他们能够概括他们的结果 - 因此这个等式可以概括,并且符号A可以代表任何2-D 线性齐次坐标变换。(这里的“齐次”仅表示变换将原点O映射到新框架的原点O′ ,对于张量来说你需要这一点,因为这样张量方程(例如ab = 0) - 或平坦度条件,黎曼张量 = 0 - 将保持不变。)并且从我们已经看到的向量和矩阵来看,这个等式还表明我们可以轻松地从原始的 2-D 旋转转移到任意维数的变换。

As we saw in chapter 1, when algebraists began to use symbols instead of words or specific numerical examples, they were able to generalise their results—so this equation can be generalised, and the symbol A can stand for any 2-D linear homogeneous coordinate transformation. (“Homogeneous” here just means that the transformation maps the origin O to the origin O′ of the new frame, and you need this for tensors, because then tensor equations such as ab = 0—or the condition for flatness, Riemann tensor = 0— will remain invariant.) And from what we’ve already seen of vectors and matrices, this equation also suggests we can easily move from our original 2-D rotation to transformations in any number of dimensions.

不同的作者使用不同的符号,但在梳理和概括坐标变换的行为方式时,我将采用当今张量分析教科书中广泛使用的符号。它与 Ricci 的符号只有细微的差别;具体来说,逆变向量分量坐标都用上层指标表示,但 Ricci 表示的坐标与我们通常在数学课上遇到的一样:x 1x 2,因此,为了进一步探索坐标旋转的结构——并了解坐标变换的一般工作原理——首先,我将使用x 1x 2表示原始坐标(x,y),使用x 1′x 2′表示新坐标(x′,y′)。然后原始旋转变换方程x′ = x cos θ + y sin θ 可以概括如下:

Different authors use different symbols, but in teasing out and generalising the way coordinate transformations behave, I’ll adapt the notation widely used in textbooks on tensor analysis today. It differs only slightly from Ricci’s; in particular, both contravariant vector components and coordinates are written with upstairs indices, but Ricci denoted coordinates as we usually meet them in maths classes: x1, x2, So, to explore a bit further the structure of a coordinate rotation—and to get a handle on how coordinate transformations work in general—first up, I’ll use x1, x2 for the original coordinates (x, y), and x1′ ,x2′ for the new ones (x′, y′). Then the original rotation transformation equation x′ = x cos θ + y sin θ can be generalised like this:

1=一个111+一个212

x1=A11x1+A21x2,

在哪里一个11=余弦θ一个21=θ在我们的特定旋转案例中。(对于表示坐标变换的矩阵元素,我使用了一个11等等,而不是线性代数中的a ij符号。你很快就会明白为什么。)类似地,变换方程y′ = − x sin θ + y cos θ 可以推广为

where A11=cosθ,A21=sinθ in our specific rotation case. (For the elements of the matrix representing the coordinate transformation, I’ve used A11, and so on, rather than the aij notation of linear algebra. You’ll see why shortly.) Similarly, the transformation equation y′ = −x sin θ + y cos θ can be generalised as

2=一个121+一个222

x2=A12x1+A22x2.

这里的指标有一个模式:在每个一般方程中,相同的虚线上行指标始终出现。这意味着你可以用一个方程来表示这两个方程,其中一般上行指标 μ′(发音为“mu-dash”)被假定取值 1 和 2(依次),因为二维空间中有两个独立的坐标:

There’s a pattern in the indices here: in each of these general equations, the same dashed upstairs index appears throughout. This means you can represent these two equations in one, where the general upstairs index μ′ (pronounced “mu-dash”) is presumed to take the values 1 and 2 (each in turn), since there are two independent coordinates in 2-D space:

μ=一个1μ1+一个2μ2

xμ=A1μx1+A2μx2.

还要注意,在右侧和式的每一项中,矩阵分量的下层索引与原始坐标上的上层索引相匹配;因此,使用希腊字母 σ(西格玛)和我在第 10 章末尾标记的求和符号,可以将上述表达式简化为:

Notice, too, that in each term in the sum on the right-hand side, the downstairs index on the matrix component is matched by an upstairs one on the original coordinate; so, using the Greek letter σ (sigma) and the summation notation I flagged at the end of chapter 10, you can simplify the above expression to:

μ=σ=12一个σμσ

xμ=σ=12Aσμxσ.

一旦爱因斯坦掌握了这一切,他会让这个符号变得更简单。他会说,看看这个模式,注意每当你有相同的上和下指标时——在这种情况下,σ像这样出现两次——你就把所有这些项加起来。既然你也知道你在哪个维度上工作,为什么不把求和符号完全去掉,让重复的指标告诉你这是一个和:

Once Einstein gets on top of all this, he’ll make this notation even simpler. He’ll say, look at the pattern and notice that whenever you have the same up and down index—in this case, a σ appearing twice like this— you add all those terms. And since you also know what dimension you’re working in, why not leave out the summation sign altogether, and let the repeated indices tell you that this is a sum:

μ=一个σμσ

xμ=Aσμxσ.

今天,这种符号被称为爱因斯坦求和约定。

Today this notation is called the Einstein summation convention.

您可以看到我的意思:当您添加更多变量时(这样您就可以处理n维空间(黎曼流形!)),您可以使用相同的符号方程,只是现在您的指标 μ′ 和 σ 将从 1 到 n 并且您的总和隐含地不包含两个项,而是n 个项。这是n 个方程(每个 μ′ 值一个),每个方程都是n 个项的总和(每个 σ 值一个项),所有这些都封装在一个小方程中。这真是太经济了!

You can see where I’m going with this: when you add more variables— so that you’re working with an n-dimensional space (Riemann’s manifold!)—you can use the same symbolic equation, except that now your indices μ′ and σ will run from 1 to n, and your sum implicitly has not two terms but n of them. That’s n equations (one for each value of μ′), each one a sum of n terms (one term for each value of σ), all encapsulated in just one small equation. It’s brilliantly economical!

我在这里选择了 μ′ 和 σ 作为指标标签,但它们是用来表示一般指标的,因此 μ 和 σ 本身并没有什么特别之处——就像学校代数中未知数的字母x是任意选择的一样(就像我选择一个σμ表示变换矩阵分量)。因此,不要关注这里的特定字母,而要关注索引的模式。正如我们将看到的,这是经济的象征这使得以张量形式表示的物理方程非常优美。

I’ve chosen μ′ and σ for the index labels here, but they are meant to represent general indices, so there’s nothing specific about the letters μ and σ themselves—just as the letter x for the unknown in school algebra is an arbitrary choice (as was my choice of Aσμ for the transformation matrix components). So don’t focus on the specific letters here, but rather on the patterns of the indices. As we’ll see, it’s this economical symbolism that makes the equations of physics so beautifully elegant when written in tensor form.

不变性和张量

INVARIANCE AND TENSORS

现在我们要了解这一切与不变性有什么关系。向量a的分量是从坐标轴测量的,因此它们以相同的方式变换,一个μ=一个σμ一个σ。例如,对于图 9.1中的旋转,分量将像上面方程中的坐标一样变换,因此,我们使用 Ricci 楼上关于(逆变)向量ab 的指标—

Now we’re getting to what all this has to do with invariance. The components of a vector a are measured from the coordinate axes, so they transform in the same way, aμ=Aσμaσ. For example, for the rotation in figure 9.1, the components will transform just like the coordinates in the equations above, so we have—using Ricci’s upstairs indices for the (contravariant) vectors a and b

a 1′ = a 1 cos θ + a 2 sin θ, a 2′ = − a 1 sin θ + a 2 cos θ,

a1′ = a1 cos θ + a2 sin θ, a2′ = −a1 sin θ + a2 cos θ,

对于b 1′b 2′也类似。当你将这些分量相乘以在旋转框架中形成标量积时,你会发现

and similarly for b1′ and b2′. When you multiply these pairs of components to form the scalar product in the rotated frame, you find that

a 1′ b 1′ + a 2′ b 2′ = a 1 b 1 + a 2 b 2

a1′b1′ + a2′b2′ = a1b1 + a2b2.

在两个框架中,你会得到相同的数字——相同的标量积——这就是不变的含义。所以这是图 9.1中几何论证的代数版本。

You get the same number—the same scalar product—in both frames, which is what it means to be invariant. So this is an algebraic version of the geometric argument in figure 9.1.

但是,当我们用本科符号表示法写出标量积a 1 b 1 + a 2 b 2 时,情况又如何呢?

But what about the scalar product as we write it in undergrad notation, a1b1 + a2b2?

在 Ricci 的符号中,这将是协变或行向量(或一元形式)的标量积。正如我们稍后会看到的,事实证明,在通常的欧几里得空间和笛卡尔坐标系中,向量分量上的上层和下层索引是没有必要区分的。但首先我们需要更多地了解下层索引对张量的意义。

In Ricci’s notation this would be the scalar product of covariant or row vectors (or one-forms). As we’ll see later, it turns out that in the usual Euclidean space and Cartesian coordinate system, there’s no need to distinguish between upstairs and downstairs indices on vector components. But first we need to see more about what the downstairs indices mean for tensors.

组件矩阵一个σμ坐标和(逆变)向量的变换显示了如何用旧坐标或向量分量来表示新坐标或向量分量。因此,要用新坐标表示原始坐标,变换则反过来。我们在图 9.3中看到了洛伦兹变换的这种变换,但你可以通过返回一般的写法来了解如何对任何坐标变换进行这种变换变换方程为X′ = A X。然后矩阵代数告诉你,要进行另一种变换,你将得到

The matrix of components Aσμ for transformations of coordinates and (contravariant) vectors shows how to write the new coordinates or vector components in terms of the old ones. So, to write your original coordinates in terms of the new ones, the transformation goes the other way. We saw this for the Lorentz transformations in figure 9.3, but you can see how to do it for any coordinate transformation by going back to writing general transformation equations as X′ = AX. Then matrix algebra tells you that to transform the other way, you’ll have

X′ = A X A −1 X′ = X

X′ = AXA−1X′ = X.

例如,旋转矩阵的逆余弦θθθ余弦θ余弦θθθ余弦θ,因为AA −1 = I。这意味着原始矩阵的第一列成为逆矩阵的第一行,第二行也是如此。换句话说,原始矩阵的列(逆变向量及其上层索引)在逆矩阵中成为行,或协变向量及其下层索引。这意味着,要表示逆矩阵的分量,您只需交换原始矩阵上的索引:一个σμ一个μσ这就是为什么 Ricci 写协变向量(单形式或对偶向量)时使用下层指标:其分量的变换规则为一个μ=一个μσ一个σ利用这条规则,你可以证明a 1 b 1 + a 2 b 2是不变的,就像a 1 b 1 + a 2 b 2是不变的一样。

For example, the inverse of the rotation matrix cosθsinθsinθcosθ is cosθsinθsinθcosθ, since AA−1 = I. This suggests that the first column in the original matrix becomes the first row in the inverse, and similarly for the second row. In other words, the columns of the original matrix—the contravariant vectors, with their upstairs indices—become rows, or covariant vectors with downstairs indices, in the inverse matrix. Which means that all you have to do to represent the components of the inverse matrix is to interchange the indices on the original one: AσμAμσ. This is why Ricci wrote covariant vectors (one-forms or dual vectors) with a downstairs index: the transformation rule for their components is aμ=Aμσaσ. Using this rule, you can prove that a1b1 + a2b2 is invariant, just as a1b1 + a2b2 is invariant.

Ricci 将向量推广到张量,他说,如果张量的所有分量的指标都在上层,无论其阶数是多少,都称为“逆变”;它将像逆变向量分量一样变换,但具有适当数量的变换矩阵一个μσ(如尾注19所示)。如果所有指标都在楼下,Ricci 称其为“协变”张量。如果一些指标在上,一些指标在下,则称为“混合”。例如,我们看到列向量与行向量的张量积的分量为u 1 v 1u 1 v 2u 2 v 1u 2 v 2等等,因此它们是混合张量的分量。

Generalising vectors to tensors, Ricci said that if all the indices on the components of a tensor, of any rank, are upstairs, it is called “contravariant”; it will transform just like contravariant vector components, but with the appropriate number of transformation matrices Aμσ (as you can see in the endnote19). If all the indices are downstairs, Ricci called it a “covariant” tensor. If some indices are up and some down, it’s called “mixed.” For example, we saw that the components of the tensor product of a column vector by a row vector were u1v1, u1v2, u2v1, u2v2, and so on, so they are the components of a mixed tensor.

这就是为什么 Ricci 说向量是张量:它们的分量(如高阶张量分量)在坐标变换下以特定方式变换。标量是张量,因为它们只是数字或数值表达式,根本不依赖于坐标,因此它们在坐标变换下自动不变。(只是点 i,并非所有数字都是不变量或标量——例如,频率取决于观察者相对运动的变化,下一章中我们将看到的多普勒效应就是一个例证。如果最终检测到了 Unruh 效应,那么温度甚至可​​能并不完全是我在第 7 章和图 11.1中所说的与坐标无关的标量——尽管你必须以接近光速的速度行进才能检测到一度的温度变化。20 )

This is why Ricci said that vectors are tensors: their components—like higher-order tensor components—transform in a specific way under a change of coordinates. And scalars are tensors because they are just numbers or numerical expressions that don’t depend on the coordinates at all—so they are automatically invariant under coordinate transformations. ( Just to dot i’s, not all numbers are invariants or scalars—for instance, frequency depends on the relative motion of the observer, as exemplified in the Doppler effect that we’ll see in the next chapter. And if the Unruh effect is finally detected, it may even turn out that temperature is not exactly the coordinateindependent scalar I said it was in chap. 7 and fig. 11.1—although you’d have to be traveling close to the speed of light to detect one degree of temperature change.20)

现代观点

A MODERN VIEW

不过,正如我们将在下一节中看到的那样,索引符号并不完全是关于坐标变换的。所以在这里我想概述一下张量的现代观点。坐标变换仍然是它的核心,但现代数学家不是通过张量的分量在这些坐标变化下的变换方式来定义张量,而是将“整个张量”定义为产生不变量的线性算子

Index notation isn’t all about coordinate transformations, though, as we’ll see in the next section. So here I want to outline a modern view of tensors. Coordinate transformations are still at the heart of it, but rather than defining tensors through the way their components transform under these coordinate changes, modern mathematicians define “whole tensors” as linear operators that yield invariants.

我们在第 6 章中看到dd是一个运算符,它的向量扩展 nabla 也是,=++。你必须在操作符中“插入”一个函数dd得到它的导数,而将函数f插入 nabla 中会得到“grad f ”,即一个向量,其分量是函数的偏导数。但张量更像散度算子 ∇∙,它对向量进行运算以得到标量。正如我们刚刚看到的,标量总是不变的。例如,在麦克斯韦方程中,∇∙ 对电场和磁场矢量进行运算,得到一个标量:

We saw in chapter 6 that ddx is an operator and so is its vector extension nabla, =xi+yj+zk.. You have to “insert” a function into the operator ddx to get its derivative, while inserting a function f into nabla gives you “grad f,” a vector whose components are partial derivatives of the function. But a tensor is more like the divergence operator, ∇∙, which operates on a vector to give a scalar. And scalars, as we just saw, are always invariant. For example, in Maxwell’s equations, ∇∙ operates on the electric and magnetic field vectors, giving a scalar:

∇ ∙ E = 4πρ

∇ ∙ B = 0。

∇ ∙ E = 4πρ

∇ ∙ B = 0.

类似地,将行向量(协变向量或单形式或对偶向量)乘以列向量(逆变向量)会得到一个标量——欧几里得空间中的标量积。因此,你可以将单形式视为向量进行运算以得到标量(不变量)的东西。这个定义直指张量的核心:不是分量和坐标变换本身但在这些变换下不变。

Similarly, multiplying a row vector (a covariant vector or one-form or dual vector) by a column vector (a contravariant vector) gives a scalar— the scalar product in Euclidean space. So, you can think of a one-form as something that operates on a vector to give a scalar, an invariant. This definition gets right to the heart of what tensors are all about: not component and coordinate transformations per se but invariance under those transformations.

上升一个阶数(或等级),你可以将矩阵视为一个混合二阶张量,它对一个向量一个一阶张量进行运算,得到一个标量。更具体地说,它对一个(列)向量进行运算,得到另一个(列)向量——就像变换方程X′ = A X中的旋转矩阵A,其中A “对”=给出一个新的向量,;然后,正如我们刚刚看到的,当你用行向量(单形式)对这个新向量进行运算时,你会得到标量积。张量的阶数越高,它必须对越多的向量和/或单形式进行运算才能得到标量。

Going up an order (or rank), you can think of a matrix as a mixed second-order tensor that operates on a vector and a one-form to give a scalar. More specifically, it operates on a (column) vector to give another (column) vector—like the rotation matrix A in the transformation equation X′ = AX, where A “operated on” X=xy to give a new vector, xy; then, as we just saw, when you operate on this new vector with a row vector (one-form), you get the scalar product. The higher the order of the tensor, the more vectors and/or one-forms it must operate on to give a scalar.

例如,张量作为算符的概念是量子理论的基础,而纯数学家将这一概念带入了更抽象的多线性映射领域。因此,关于线性算符的概念,以及“向量空间”的概念,以及其他微妙之处,如向量和一元形式之间的差异,以及转换“基”向量和一元形式而不是向量和张量分量的意义,还有很多话要说,但这超出了我的范围。不过,如果您到目前为止一直跟着我,我希望本节和上一节能让您感受到数学家发展思想的方式——他们如何整理他们的数学构造必须遵循的规则,以及他们如何根据不变性等重要思想来解释这些规则和构造。随着数学家在前人见解的基础上不断发展,这些解释也在不断发展。这是本书的一个关键主题,我们已经看到了这种数学发展,从楔形文字表和计算算法到符号代数、向量和矩阵,从微积分到矢量分析。在本章后面,我们将迈出张量微积分的最后一步。

This conception of tensors as operators is fundamental in quantum theory, for example, while pure mathematicians take the idea into the more abstract territory of multilinear mappings. So there’s much more to say about the notion of linear operators—and about “vector spaces,” too, and other subtleties such as the difference between vectors and one-forms, and the significance of transforming “basis” vectors and one-forms rather than vector and tensor components—but that’s beyond my scope here. Still, if you’ve stayed with me so far, I hope this section and the previous one have given you a feeling for the way mathematicians develop their ideas—how they sort out the rules that their mathematical constructs have to obey, and how they interpret these rules and constructs in terms of important ideas such as invariance. These interpretations evolve as mathematicians build on their forerunners’ insights. This is a key theme of this book, where we’ve already seen this kind of mathematical development, from cuneiform tables and computational algorithms to symbolic algebra, vectors, and matrices, and from calculus to vector analysis. Later in this chapter we’ll take the final step to tensor calculus.

张量符号的惊人计算能力

THE AMAZING COMPUTATIONAL POWER OF TENSOR SYMBOLISM

张量分量上的索引位置在张量方程和计算中起着至关重要的作用。例如,我们已经看到,在欧几里得空间中,将行(协变)向量v乘以列(逆变)向量u可得出它们的标量积。使用 Ricci 的指数符号和爱因斯坦的求和约定,我们可以得到一个非常经济的表示:

The position of the indices on tensor components plays a vital role in tensor equations and computations. For example, we’ve seen that in Euclidean space, multiplying a row (covariant) vector v by a column (contravariant) vector u gives their scalar product. Using Ricci’s index notation and Einstein’s summation convention we have a beautifully economical representation of this:

v 1 u 1 + v 2 u 2 == v μ u μ

v1u1 + v2u2vμuμ.

我在这里选择字母 μ 并没有什么特别之处——同样,我可以选择任何字母,因为重要的是两个索引相同(所以这是一个和)。21这种表示法的神奇之处在于:每对上和下索引都相同,这实际上告诉我们,这个标量积在适当的坐标变换下是不变的。您可以通过插入变换方程来证明它是不变的,正如您在下一个尾注中看到的那样。但是一旦您理解了如何做到这一点,索引符号就可以省去您的麻烦。22

There’s nothing special about my choice of the letter μ here—again, I could have chosen any letter, because what matters is that both indices are the same (so this is a sum).21 The amazing thing about this representation is this: the fact that each pair of up and down indices is the same actually tells us that this scalar product is invariant under appropriate coordinate transformations. You can prove it’s invariant, simply by inserting the transformation equations, as you can see in the next endnote. But once you understand how to do that, the index notation saves you the bother.22

张量符号让事情变得简单,这确实很了不起。不过,你可能会抱怨,到目前为止,我们已经有了三种不同类型的标量积,即上层指标、下层指标,现在还有混合指标。这只是因为在张量分析中有两种类型的向量,但我将很快展示它们是如何在一个通用表达式中组合在一起的。现在,我想重点介绍标量积的混合形式通过其符号本身表现出不变性的非凡方式。这也适用于高阶张量,因此每当你看到一个表达式,其中每个下层指标都与相同的上层指标相匹配时——例如T μν h μν(ν 发音为“nu”)——你就知道它是不变的。这真的很了不起——这只是里奇的指标符号如此出色的创新的一个例子。他说得对,这值得为之奋斗。

It really is remarkable the way tensor notation makes things easier. Nonetheless, you might be grumbling that so far, we’ve had three different types of scalar product, with upstairs indices, downstairs indices, and now mixed indices. That’s simply because in tensor analysis there are two types of vector, but I’ll show shortly how it all comes together in one general expression. For now, I want to focus on the remarkable way the mixed form of the scalar product shows invariance through its very symbolism. This carries over for higher-order tensors, too, so whenever you see an expression where each downstairs index is matched by the same upstairs one—such as Tμνhμν (ν is pronounced “nu”)—you know it is invariant. It’s quite extraordinary, really—and this is just one example of why Ricci’s index notation is such a brilliant innovation. He was right to say it is worth the struggle of initiation.

诸如v μ u μT μν h μν 之类的张量表达式是称为“收缩”的张量运算的示例 - 因为当您将一对楼上和楼下的指标设置为相等时,您正在减少或收缩张量的秩。例如,v μ u λ(其中 λ 发音为“lambda”)是混合秩 2(双指标)张量的一般分量,但是当您设置 λ = μ 时,您会将秩(或阶数)降低为 0,因为v μ u μ是标量(标量积)。

Tensor expressions such as vμuμ and Tμνhμν are examples of the tensor operation called “contraction”—because when you set a pair of upstairs and downstairs indices equal, you’re reducing, or contracting, the rank of your tensor. For instance, vμuλ (where λ is pronounced “lambda”) is a general component of a mixed rank 2 (two-index) tensor, but when you set λ = μ, you reduce the rank (or order) to 0, because vμuμ is a scalar (the scalar product).

除非您想找到不变量,否则不必收缩所有指标。例如,T μν h λσ是 4 阶(4 指标)张量的一般分量,但如果您设置 λ = μ ,则会得到一个双指标张量,其分量为T μν h μσ。它是一个 2 指标张量,因为您在重复指标 μ 上求和,只留下 ν 和 σ 指标自由。这就像收缩 μ 指标“取消”它们一样,就像您在链式法则中“取消”项的方式一样,dd=dddd,尽管在这种情况下您是在对项求和而不是“删除”它们。

You don’t have to contract all the indices unless you want to find the invariants. For instance, Tμνhλσ is a general component of a rank 4 (4-index) tensor, but if you set λ = μ you get a two-index tensor, with components Tμνhμσ. It’s a 2-index tensor because you’re summing on the repeated index μ, leaving only the ν and σ indices free. It’s as if contracting the μ indices “cancels” them, rather like the way you “cancel” terms in the chain rule, dydx=dydududx, although in this case you’re summing terms rather than “deleting” them.

有一个特别重要的缩略词现在被称为“内积”,以纪念格拉斯曼。如前所述,哈密顿的系统演变成我们大学水平的矢量分析,它非常适合三维问题——还记得那些让他走上发现四元数之路的三维旋转吗!格拉斯曼的更抽象,所以虽然它更难应用,但它更容易适用于黎曼创建的n维空间,而里奇的张量就是在这些空间中运算的。因此,到了 20 世纪初,格拉斯曼的思想开始渗透到主流,为源自哈密顿的矢量和张量分析增加了概念实质。里奇遵循哈密顿传统,在他 1900 年的微积分概述中,他没有使用“内积”(“外积”)这个术语;然而,到 1916 年——仅举一个例子——爱因斯坦在他的广义相对论概述中使用了这些格拉斯曼术语。

There’s an especially important contraction now called the “inner product” in tribute to Grassmann. As we saw earlier, Hamilton’s system, which morphed into our university-level vector analysis, was perfectly adapted for three-dimensional problems—remember those 3-D rotations that had set him on the path to discovering quaternions! Grassmann’s was more abstract, so although it was harder to apply, it was more readily adaptable to the n-dimensional spaces that Riemann created, and in which Ricci’s tensors operate. So, by the early twentieth century, Grassmann’s ideas had begun filtering into the mainstream, giving added conceptual substance to the vector and tensor analysis that had descended from Hamilton. Ricci was in the Hamiltonian tradition, and in his 1900 overview of his calculus he doesn’t use the term “inner” (or “outer”) product; by 1916, however— and to take just one example—in his overview of general relativity theory Einstein will use these Grassmannian terms.

那么,什么是内积?它来自于收缩由两个张量的外积形成的混合张量的一对指标。例如,假设你形成一个协变张量T(分量为T μν)和一个向量u(分量为u σ)的外积。你得到一个新的混合张量,其分量为T μν u σ。现在通过设置 σ = μ 收缩指标,得到T μν u μ ;这是Tu的内积的一般分量。(以分量形式表示,它是T u 1 + T u 2 + … ,项数取决于空间的维数。)

So, what is the inner product? It comes from contracting a pair of indices on a mixed tensor formed from the outer product of two other tensors. For example, suppose you form the outer product of a covariant tensor T with components Tμν and a vector u with components uσ. You get a new mixed tensor whose components are Tμνuσ. Now contract the indices by setting σ = μ, to give Tμνuμ; this is the general component of the inner product of T and u. (In component form, it is Tu1 + Tu2 + … , the number of terms depending on the dimension of the space.)

但是,如果你把这个张量与另一个(逆变)向量v相乘,会发生什么情况呢:你会得到一个新的张量,其分量为T μν u μ v λ。(因此,内积会降低或收缩秩,而外积积会增加它。)如果现在设置 λ = ν,则会得到另一个新张量,其分量为T μν u μ v ν 。这是T μν u μv λ的内积。与标量积v μ u μ一样,此张量是标量(不变数或函数),因为每对指标都相同。

But look what happens if you take the outer product of this tensor with another (contravariant) vector, v: you get a new tensor, with components Tμνuμvλ. (So inner products reduce or contract the rank, and outer products increase it.) If you now set λ = ν, you get yet another new tensor, with components Tμνuμvν. This is the inner product of Tμνuμ and vλ. Like the scalar product vμuμ, this tensor is a scalar (an invariant number or function), because each pair of indices is the same.

事实上,如果T是一个度量张量(从现在开始,我将引用爱因斯坦的公式,用g表示,分量为g μν),那么这个特定的内积实际上就是我们习惯称之为uv的标量积。因为,正如我们所见,度量实际上定义了标量积。例如,二维欧几里得度量

In fact, if T is a metric tensor—which from now on, and cribbing from Einstein, I’ll denote by g, with components gμν—then this particular inner product is, in fact, just what we’ve been used to calling the scalar product of u and v. For, as we’ve seen, the metric actually defines the scalar product. For example, the 2-D Euclidean metric

ds 2 = dx 2 + dy 2 ≡ ( dx 1 ) 2 + ( dx 2 ) 2

ds2 = dx2 + dy2 ≡ (dx1)2 + (dx2)2

有分量g 11 = g 22 = 1,其他分量为零;因此,写出重复指标所表示的和,此例中的内积为:

has components g11 = g22 = 1, with the other components zero; so, writing out the sums indicated by the repeated indices, the inner product in this case is:

g μν u μ v ν = g 11 u 1 v 1 + g 12 u 1 v 2 + g 21 u 2 v 1 + g 22 u 2 v 2 = u 1 v 1 + u 2 v 2

gμνuμvν = g11u1v1 + g12u1v2 + g21u2v1 + g22u2v2 = u1v1 + u2v2.

这实际上是通常的向量分析标量积uv,只是带有 Ricci 楼上的指标,因为这里两个向量都是逆变的。

This is, indeed, the usual vector analysis scalar product uv, except with Ricci’s upstairs indices because here both vectors are contravariant.

了解对称性、为什么度量是张量,以及对标量积的指标进行排序

A PEEK AT SYMMETRY, WHY METRICS ARE TENSORS, AND SORTING OUT THE INDICES ON SCALAR PRODUCTS

我们在第 4 章中看到,标量积是可交换的(只有矢量积不是)。并且我们刚刚看到uv = g μν u μ v ν(并且蕴涵vu = g νμ v ν u μ)。所以,这个交换性uv = vu意味着我们必须有g μν = g νμ。因此,度量张量分量上的指标是对称的,就像镜子中的反射一样。这种对称性在线性坐标变换下是不变的,因为标量积是不变的,所以它在计算中很方便。正如我们所见,不变性通常是一种数学“对称性”,因为当某物不变时,它会保持不变 - 就像反射图像或旋转的雪花的形状一样。

We saw in chapter 4 that the scalar product is commutative (it’s only the vector product that isn’t). And we just saw that uv = gμνuμvν (and by implication vu = gνμvνuμ). So, this commutativity, uv = vu, means that we must have gμν = gνμ. The indices on the metric tensor components are, therefore, symmetric, like a reflection in a mirror. This symmetry is invariant under linear coordinate transformations because the scalar product is, so it’s handy in computations. As we’ve seen, invariance in general is a mathematical “symmetry,” because when something is invariant, it stays the same—just like the shape of a reflected image or a rotated snowflake.

在第 9 章中,我们介绍了欧几里得度量和闵可夫斯基度量,其中微分的系数是常数,这意味着它们定义了平坦的空间。在第 10 章中,我们看到高斯证明了曲面曲率的信息包含在一般二维度量的微分系数中,而黎曼将此推广到弯曲的n维空间。因此,对于任意坐标x μ,弯曲空间中的度量可以表示为

In chapter 9 we met the Euclidean and Minkowski metrics, where the coefficients of the differentials are constant, signifying that they define flat spaces. In chapter 10, we saw that Gauss proved information about the curvature of a surface is contained in the coefficients of the differentials in the general 2-D metric, and that Riemann generalised this to curved n-dimensional spaces. So, with arbitrary coordinates xμ, a metric in curved space can be expressed as

ds 2 = g μν dx μ dx ν

ds2 = gμνdxμdxν,

其中系数g μν不是常数而是坐标的函数。

where now the coefficients gμν are not constants but functions of the coordinates.

从重复的指标中,你可以直接看出“距离”或时空间隔量度量ds 2是不变的,你可以使用坐标变换方程来证明这一点。23我们已经看到了这个一般结果的例子:欧几里得度量在旋转等变换下是不变的,而闵可夫斯基度量在洛伦兹变换下是不变的(图 9.3)。但为什么度量是张量呢?线索就在不变性中;毕竟,表示不变性是张量的全部意义所在。

From the repeated indices you can see straightaway that the “distance” or space-time interval measure, ds2, is invariant, and you can prove it by using the coordinate transformation equations.23 We’ve already seen examples of this general result: the Euclidean metric is invariant under transformations such as rotations, while the Minkowski metric is invariant under Lorentz transformations (fig. 9.3). But why is the metric a tensor? The clue is in the invariance; after all, representing invariance is the whole point of tensors.

为了更详细地说明这一点,我们之前已经看到,g μν u μ v ν是向量uv的标量积。这表明度量g “作用于”这两个向量以产生一个标量——不变的标量积。这意味着度量是一个张量,符合我在前两节中给出的现代定义。

To see this in more detail, we saw just before that gμνuμvν is the scalar product of the vectors u and v. This suggests that the metric g “operates on” these two vectors to produce a scalar—the invariant scalar product. Which means the metric is a tensor according to the modern definition I gave two sections back.

根据里奇的定义,它也是一个张量,因为我们知道逆变向量分量u μ , v ν 的变换方程,所以如果g μν u μ v ν是一个不变标量,那么g μν的变换方程必须是协变秩为 2 的张量的变换方程。前面的尾注说明了这一点的计算,但您已经可以看到现代观点更加优雅。

It’s also a tensor according to Ricci’s definition, because we know the transformation equations for the contravariant vector components uμ, vν, so the transformation equation of gμν must be that of a covariant rank 2 tensor if gμνuμvν is to be an invariant scalar. The previous endnote illustrates the calculations that show this, but you can see already that the modern view is more elegant.

• • •

• • •

在简要介绍里奇的最高成就——张量导数之前,我还有最后一件事想向你们展示。当我之前谈到标量积的各种形式v 1 u 1 + v 2 u 2v 1 u 1 + v 2 u 2v 1 u 1 + v 2 u 2 时我假设我们在二维欧氏空间中,其中度量是ds 2 = dx 2 + dy 2。更一般地,我们刚刚看到,在一个度量分量为g μν的空间中,两个(逆变)向量vu的标量积是g μν u μ v ν。现在看看如果我交换指标的位置并写成g μν v μ u ν会发生什么。这表明两个协变向量的标量积。但g μν是什么?Ricci 定义g μν使其具有非常特殊的属性:它“提高”协变张量的指标。我们之前看到,具有重复(收缩)μ 指标的T μν h μσ是一个双指标张量,就好像我们在对它们求和时“取消”了 μ 一样。因此,Ricci 定义了特殊的收缩

There’s one last thing I want to show you before I briefly outline Ricci’s crowning achievement, tensor derivatives. When I spoke earlier of the scalar product in the various forms v1u1 + v2u2, v1u1 + v2u2, and v1u1 + v2u2, I was assuming we were in 2-D Euclidean space, where the metric is ds2 = dx2 + dy2. More generally, we’ve just seen that in a space with a metric whose components are gμν, the scalar product of two (contravariant) vectors v and u is gμνuμvν. Now look what happens if I swap the positions of the indices and write g μνvμuν. This suggests the scalar product of two covariant vectors. But what is g μν? Ricci defined g μν so that it has a very special property: it “raises the index” of a covariant tensor. We saw a little earlier that Tμνhμσ, with repeated (contracted) μ indices, is a two-index tensor, as if we’d “canceled” the μ’s when we summed them. So, Ricci defined the special contractions

μμσ=σλσσ=λ

gμvgμσ=gσv,andgλσgσv=gλv.

(实际上,Ricci 使用的是a rs而不是g μν,但除此之外,我使用了他的定义。)换句话说,g μν已将g μσ带到σ。而g μν g λσ已将g μσ变为g λν。反之亦然:g μν可以降低指标。(这是因为 Ricci 本质上将度量分量g μνg μν的矩阵表示定义为彼此的逆。但关键是这些定义。)

(Actually, Ricci used ars rather than g μν, but otherwise I’m using his definitions.) In other words, g μν has taken gμσ to gσv. And gμνgλσ has taken gμσ to gλν. It works the other way, too: gμν can lower the indices. (That’s because Ricci essentially defined the matrix representations of the metric components gμν and g μν to be inverses of each other. But the key thing is that these are definitions.)

里奇将此定义为一种普遍性质,因此度量张量在与任何张量收缩时都可以提高或降低指标。正如我们将看到的,这在爱因斯坦等张量方程中非常重要。但它也以一种相当巧妙的方式将所有这些不同形式的标量积结合在一起。收缩g μν v μ降低了向量的指标,得到v ν。这意味着,一般而言,g μν v μ u ν = v ν u ν,就像我之前对行和列向量的特定标量积所说的那样!类似地,g μν v μ u ν = v ν u ν

Ricci defined this as a general property, so that the metric tensor can raise or lower indices when it is contracted with any tensor. This is important in tensor equations such as Einstein’s, as we’ll see. But it also brings all those different forms of the scalar product together in a rather brilliant way. The contraction gμνvμ lowers the index on the vector, giving vν. This means that in general, gμνvμuν = vνuν, just as I had earlier for the particular scalar product of the row and column vector! Similarly, g μνvμuν = vνuν.

但真正有趣的是,在欧几里得空间中,使用笛卡尔坐标,我们知道度量是ds 2 = dx 2 + dy 2。为了简单起见,我保留 2-D,但当然你可以添加更多维度,而我要得到的结果将是相同的:也就是说,在这种情况下,无论你用楼上索引还是楼下索引来写向量的索引都没有关系。这是因为(在 2-D 中)

But here’s the really interesting thing. In Euclidean space, using Cartesian coordinates, we know that the metric is ds2 = dx2 + dy2. I’m keeping to 2-D for simplicity, but of course you can add more dimensions and the result I’m heading to will be the same: namely, in this situation it doesn’t matter if you write your vectors’ indices with upstairs indices or downstairs ones. That’s because (in 2-D)

v ν = g μν v μ = g v 1 + g v 2 ,

vν = gμνvμ = gv1 + gv2,

但欧氏度量的唯一非零分量是g 11 = 1 = g 22,因此,有

but the only nonzero components of the Euclidean metric are g11 = 1 = g22, so, you have

v 1 = g 11 v 1 + g 21 v 2 = 1 × v 1 + 0 × v 2 = v 1 ,

v1 = g11v1 + g21v2 = 1 × v1 + 0 × v2 = v1,

类似地,v 2 = g 12 v 1 + g 22 v 2 = v 2。换句话说,在具有通常笛卡尔坐标的欧几里得空间中,向量的分量和一元形式之间没有区别(即里奇的逆变向量和协变向量之间没有区别)。这就是为什么在普通的向量分析中,无需担心术语或指标位置上的这种区别。

and similarly, v2 = g12v1 + g22v2 = v2. In other words, in Euclidean space with the usual Cartesian coordinates, there’s no difference between components of a vector and a one-form (that is, between Ricci’s contravariant and covariant vectors). That’s why, in ordinary vector analysis, there’s no need to worry about this distinction in terminology or in the position of indices.

张量微积分简介

TENSOR CALCULUS, VERY BRIEFLY

对于里奇来说,最大的问题是,如果对张量进行微分会发生什么?更具体地说,导数是张量吗?如果不是,那么张量在物理学中用处不大,因为物理现象通常用微分方程来建模——例如牛顿运动定律和麦克斯韦电磁方程。正如我们在第 9 章中看到的那样,即使从一个参考系切换到另一个参考系,物理方程也必须保持其形式不变。否则,不同的观察者会推导出不同的物理定律,我们将永远无法就物理现实的本质达成一致。

The biggest question for Ricci was, what happens if you differentiate a tensor? More specifically, is the derivative a tensor? If it’s not, then tensors are not much use in physics, where physical phenomena are widely modeled by differential equations—as in Newton’s laws of motion and Maxwell’s equations of electromagnetism, for example. As we saw in chapter 9, the equations of physics must also keep their form invariant, even when you change from one reference frame to another. Otherwise, different observers would deduce different laws of physics, and we’d never be able to agree on the nature of physical reality.

形式不变的方程被称为“协变”方程,里奇将他的不变导数称为“协变导数”。它不同于普通导数或偏导数,因为事实证明,矢量分量u μ对该框架中某个坐标(比如x λ)的偏导数一般不会像张量那样变换。里奇利用克里斯托费尔发现的不变表达式找到了导数的正确协变形式。这相当于在偏导数中添加一个涉及我在第 10 章中提到的克里斯托费尔符号的项。里奇效仿克里斯托费尔,用花括号表示这些符号,但今天,效仿爱因斯坦等人,它们通常表示为Γσλμ(符号 Γ 是大写的希腊字母 gamma;这些指标符合相关的变换方程,尽管今天它们被解释为基向量导数的系数。例如,在笛卡尔坐标系中,基向量为i、j、k。它们是常数,因此它们的导数为零,因此在这种情况下克里斯托费尔符号也为零。因此,在欧几里得空间中,您只需要偏导数!

Form-invariant equations are called “covariant,” and Ricci called his invariant derivative a “covariant derivative.” It’s not the same as an ordinary or partial derivative, because it turns out that the partial derivative of a vector component uμ with respect to one of the coordinates in that frame, say xλ, doesn’t, in general, transform like a tensor. Ricci found the right covariant form of the derivative by using an invariant expression discovered by Christoffel. It amounts to adding to the partial derivative a term involving the Christoffel symbols I mentioned in chapter 10. Ricci followed Christoffel and denoted these symbols with curly brackets, but today, following Einstein and others, they’re usually denoted by Γσλμ. (The symbol Γ is the upper-case Greek letter gamma; the indices fit in with the relevant transformation equations, although today they are interpreted as the coefficients of the derivatives of basis vectors. For instance, in Cartesian coordinates, the basis vectors are i, j, k. They are constant, so their derivatives are zero, and hence so are the Christoffel symbols in this case. So, in Euclidean space, you only need partial derivatives!)

对向量或张量分量取偏导数会得到第二个指标——用来表示与该分量求导的变量。​​Ricci 简单地通过添加另一个指标来表示向量的偏导数,但今天,新的指标用逗号表示,以表明它是导数。因此,u μ的偏导数表示为

Taking the partial derivative of a vector or tensor component gives it a second index—to show the variable with which the component is being differentiated. Ricci represented the partial derivative of a vector simply by adding another index, but today, the new index is indicated with a comma to show it’s a derivative. So, the partial derivative of uμ is represented as

μλμλ

uμxλuμ,λ.

通常使用分号来表示协变导数:

A semicolon is often used to designate the covariant derivative:

μλ=μλ+Γμσλσ

uμ;λ=uμ,λ+Γμσλuσ.

我展示这个方程只是为了给你一个直观的印象,所以细节并不重要——只是要说,任何张量的协变导数都有自然延伸,而不仅仅是向量。关键是它是一个张量——所以它的值在相关的坐标变换下是不变的。换句话说,当你变换到新的坐标并取变换后的分量u μ′x λ′的协变导数时,你会得到相同的结果。

I’ve shown this equation just to give you a visual, so the details are not important—except to say that there’s a natural extension to covariant derivatives of any tensor, not just vectors. The key thing is that it’s a tensor— so its value is invariant under the relevant coordinate transformations. In other words, you get the same result when you transform to new coordinates and take the covariant derivative of the transformed component uμ′ with respect to xλ′.

在笛卡尔坐标系中,偏导数和协变导数之间的差异在平坦空间和时空中消失了。这是在弯曲空间中很重要的区别在平坦空间中消失的另一个例子:在我们在学校学习的欧几里得向量分析中,不需要讨论协变导数,就像不需要担心向量上指标的位置一样。

In Cartesian coordinates, the difference between partial and covariant derivatives disappears in flat spaces and space-times. This is another example of the way distinctions that are important in curved spaces disappear in flat space: in the Euclidean vector analysis we learn at school, there’s no need to talk about covariant derivatives, just as there’s no need to worry about the position of the indices on vectors.

没人关心!

NOBODY CARED!

人们多年来一直在研究不变性和微分几何,里奇将二者整合到张量分析中,即他的“绝对微分学”。他借鉴了一些早期研究人员的研究成果,主要是高斯、不仅有黎曼、克里斯托费尔,还有索福斯·李,他在群和不变性方面的工作至今仍具有重要意义,还有著名的微分几何学家欧亨尼奥·贝尔特拉米,里奇在博洛尼亚读书时曾是他的教授。

People had been studying invariance and differential geometry for years when Ricci brought it all together in tensor analysis—his “absolute differential calculus.” He drew on some of these earlier researchers—chiefly Gauss, Riemann, and Christoffel, but also others such as Sophus Lie, whose work on groups and invariance is still important, and the renowned differential geometer, Eugenio Beltrami, who’d been Ricci’s professor when he was a student at Bologna.

19 世纪 80 年代末,里奇将他的一些关于张量的论文提交给意大利皇家数学奖评委,贝尔特拉米担任评委。他代表评委发言,赞赏里奇的数学才华,但他怀疑,为创造这种新微积分所付出的努力是否会得到足够卓有成效的应用回报——这些应用是现有方法无法实现的。24他似乎对此表示怀疑——就像威廉·汤姆森和阿瑟·凯莱认为整体矢量分析与单独分量计算相比没有任何优势一样。 (就在贝尔特拉米在意大利发表关于张量的言论的同时,英国也爆发了矢量之争。)但正如麦克斯韦认为整个矢量在物理学中的重要性在于它们提供的物理洞察力一样,里奇后来写道,在理解弯曲的n维曲面和空间方面,他的微积分及其符号“不仅有助于优雅地展示,也有助于敏捷和清晰地展示和得出结论。” 25

When Ricci entered some of his papers on tensors for Italy’s Royal Mathematics Prize in the late 1880s, Beltrami was a judge. Speaking on behalf of the judging committee, he admired Ricci’s mathematical virtuosity, but he wondered if the effort that had gone into creating this new calculus would ever be repaid by sufficiently fruitful applications—applications that could not be made using existing methods.24 He rather seemed to doubt it—just as William Thomson and Arthur Cayley could see no benefit in whole-vector analysis over separate component calculations. (The vector wars were going on in Britain at the very same time as Beltrami’s pronouncement about tensors in Italy.) But just as Maxwell had argued that the importance of whole vectors in physics was the physical insight they offered, so Ricci would later write that when it came to understanding curved n-dimensional surfaces and spaces, his calculus and its notation “contribute not only to the elegance, but also to the agility and clarity of the demonstrations and conclusions.”25

然而,回想十九世纪八十年代,利玛窦当时还未晋升为教授,他一定觉得,尽管他在学生时代有着辉煌的成就,但却永远无法得到认可和实现梦想。

Back in the 1880s, though, it must have seemed to Ricci—who still hadn’t got his promotion to professor—that for all the promise of his brilliant student years, he would forever go unrecognised and unfulfilled.

(12)一切都汇聚在一起

(12) EVERYTHING COMES TOGETHER

张量和广义相对论

Tensors and the General Theory of Relativity

作为真正的开拓者,里奇从未停止相信张量微积分的重要性。后来,在最初难以察觉的情况下,它的命运开始转变。第一个征兆是,1890 年,一名出色的学生图利奥·列维-奇维塔 (Tullio Levi-Civita) 进入里奇的班学习——同年,37 岁的里奇终于获得了全职教授职位。列维-奇维塔以几乎完美的成绩从中学毕业,1894 年,他也以几乎完美的成绩从帕多瓦大学毕业。甚至他的职业生涯也以完美的轨迹开始,29 岁时,他从里奇的助手升为同事教授。

True trailblazer that he was, Ricci never stopped believing in the importance of his tensor calculus. Then, imperceptibly at first, the tide of its fortune began to turn. The first augury came when a remarkable student, Tullio Levi-Civita, enrolled in Ricci’s class in 1890—the same year thirty-seven-year-old Ricci finally won his full professorship. Levi-Civita had graduated from his secondary school with an almost perfect score in all his subjects, and in 1894 he graduated from the University of Padua with an almost perfect score, too. Even his career began with a perfect trajectory, from Ricci’s assistant to his fellow professor by the time he was twenty-nine.

热情洋溢、世俗的列维-奇维塔与他严肃、虔诚的前教授截然不同——然而,这两个人直到里奇去世前仍然是亲密的朋友和同事。他们撰写了许多联合论文,发展了里奇的“绝对微分学”,这导致了张量微积分的第二次转运——当时费利克斯·克莱因建议他们写一篇关于它的概述,将其推向主流。1900 年,克莱因在他的《数学年鉴》上发表了这篇开创性的论文——十二年后,爱因斯坦和格罗斯曼读到了它,并被震撼了。1

The ebullient, secular Levi-Civita couldn’t have been more different from his straitlaced, devout former professor—yet the two men would remain close friends and colleagues till the day Ricci died. They wrote many joint papers developing Ricci’s “absolute differential calculus,” and this led to tensor calculus’s second change of fortune—when Felix Klein suggested they write an overview of it, to put it out into the mainstream. In 1900, Klein published this seminal paper in his journal Mathematische Annalen— and twelve years later, Einstein and Grossmann read it, and were blown away.1

最快乐的想法:爱因斯坦在发现张量之前对引力的思考

THE HAPPIEST THOUGHT: EINSTEIN’S THINKING ABOUT GRAVITY BEFORE HE DISCOVERED TENSORS

我们在第 10 章中看到,1912 年秋天——爱因斯坦加入老朋友马塞尔·格罗斯曼在瑞士“理工学院”或 ETH 任教后不久——格罗斯曼建议使用里奇的“绝对”微积分将物理定律从狭义相对论框架转移到包含引力的广义理论。但为了达到这一点,爱因斯坦已经做了很多思考。

We saw in chapter 10 that in the autumn of 1912—soon after Einstein joined his old friend Marcel Grossmann on the faculty of the Swiss “Poly” or ETH—Grossmann suggested Ricci’s “absolute” calculus as the way to transfer the laws of physics from the framework of special relativity to a general theory that included gravity. But to get to that point Einstein had already done a lot of thinking.

在太空旅行出现之前,找到引力理论的困难在于,引力在地球上始终存在。对于其他力,例如电磁力,您可以根据实验建立理论,这些实验展示了一个带电粒子如何在另一个粒子的场中移动。但是您无法轻易地分离一个物体对另一个物体的引力作用,因为与地球引力场的影响相比,日常物体之间的力可以忽略不计,地球引力场具有一个显著的特性,即当没有空气阻力时,所有物体都会以相同的速率向下拉。这当然是伽利略的著名结果,匈牙利男爵罗兰·冯·厄特沃什 (Roland von Eötvös) 于 1909 年以前所未有的精度证实了这一结果,并于 2022 年通过法国卫星显微镜的观测以惊人的精度证实了这一结果,精度约为千万亿分之一。2

The difficulty in finding a theory of gravity in those days before space travel was that gravity is always with us here on Earth. With other forces, such as electromagnetism, you can build up a theory based on experiments that show how one charged particle moves in the field of another, for example. But you cannot readily isolate the gravitational effect of one body on another, because the force between everyday objects is negligible compared with the effect of Earth’s gravitational field, which has the remarkable property that when there’s no air resistance, all material bodies are drawn downward at the same rate. This of course is Galileo’s famous result, which the Hungarian Baron Roland von Eötvös confirmed with unprecedented accuracy in 1909, and which was confirmed in 2022 to a spectacular accuracy of about one part in a thousand trillion, by observations made with the French satellite MICROSCOPE.2

牛顿的引力理论就是以这一下落运动定律为基础的,并且是基于对月球和行星运动的观察而得出的巧妙推论。这一定律在太阳系的大多数应用中仍然非常准确,但在 1907 年爱因斯坦开始思考引力时,有一个众所周知的例外。每颗行星都以椭圆轨道绕太阳旋转,但由于其他行星的引力,这些椭圆轨道会缓慢地进动——就像倾斜的陀螺在地球向下的引力作用下绕其轴线进动一样。这意味着太阳和近日点(行星轨道上最靠近太阳的点)之间的连线随着行星的每次旋转而略有偏移——但就水星而言,牛顿计算的近日点进动误差仅为每世纪 43 角秒。角秒或弧秒是 1/3600 度,因此 19 世纪的天文学家竟然能够检测到这种微小的差异,这真是令人惊叹。3那么,如何创建更精确的引力理论呢?

Newton had based his theory of gravity on this law of falling motion— and on ingenious deductions from the observed motion of the moon and planets. It is still extremely accurate for most applications within the solar system, but in 1907 when Einstein began thinking about gravity, there was a well-known exception. Each planet revolves around the Sun in an elliptical orbit, but because of the gravitational pull of the other planets, these elliptical orbits slowly precess—the way a tilted spinning top precesses around its axis because of the downward pull of Earth. This means that the line between the Sun and the perihelion (the point in the planet’s orbit that is closest to the Sun) shifts slightly with each planetary revolution— but in the case of Mercury, the Newtonian calculation for the precession of the perihelion was out by a tiny 43 angular seconds a century. An angular second, or arc second, is 1/3600th of a degree, so it is amazing that nineteenth-century astronomers had managed to detect this miniscule difference.3 So, how to create a more accurate theory of gravity?

爱因斯坦还在专利局工作时,突然有了奇妙的领悟——“这是我一生中最快乐的想法”。4意识到,事实上,我们在地球上可以富有想象力地“消除”地球引力的影响,只需改变我们的参考系。我们通常的视角是从一个牢牢固定在地面的框架,但如果我们是自由落体——比如从屋顶上掉下来——会怎么样?想象你处于这种眩晕的情形,然后假设你放开了手里的一个球。从地面观看的人会看到你和球都以相同的速度继续下落。这就是我们从陆地上看下落物体的行为。然而,相对于你而言,下落的球保持静止——就好像根本没有引力作用在它身上一样。

Einstein was still at the patent office when he had a marvelous insight— “the happiest thought of my life.”4 He realised that we on Earth can, in fact, imaginatively “transform away” the effect of Earth’s gravity, simply by changing our frame of reference. Our usual point of view is from a frame fixed firmly to the ground, but what if we are in free fall—falling off a roof, for example? Imagine that you are in this vertiginous situation, and then suppose you let go of a ball you were holding. A person watching from the ground will see that both you and the ball continue to fall at the same rate. That is how falling objects behave when we view them from terra firma. Relative to you, however, the dropped ball stays motionless—just as if there were no gravity acting on it at all.

换句话说,你可以利用自由落体的观察者(比如从屋顶掉下来的可怜人)来“消除”引力场——爱因斯坦不知道在宇宙飞船中“自由落体”更安全!但这还不是全部:爱因斯坦意识到你也可以“制造”引力场——通过将自己置于向上加速的框架中,例如电梯。我们将在图 12.1中探讨这个想法,但首先,爱因斯坦从他的“快乐想法”中得到的是,他确实可以设计出一个相对论引力理论,因为我们可以通过简单地改变参考系来制造或消除引力场。 (但是,我们只能“局部地”做到这一点。5 这就是为什么 2022 年 MICROSCOPE 合作项目的结果如此重要:如果所有物体在相同的引力作用下不以相同的速率下落,那么自由落体的观察者就无法“消除”引力,整个广义相对论的概念就会崩溃。

In other words, you could “unmake” a gravitational field by using a free-falling observer like the poor person falling off the roof—Einstein didn’t know that you could “free fall” more safely in a spaceship! But that’s not all: Einstein realised that you could “make” a gravitational field, too—by putting yourself in an upwardly accelerating frame such as an elevator. We’ll explore this idea in figure 12.1, but first, what Einstein took from his “happy thought” was that he could indeed devise a relativistic theory of gravity, since we can make or unmake a gravitational field simply by changing reference frames. (We can only do this “locally,” though.5) This is why results such as those of the 2022 MICROSCOPE collaboration are so crucial: if all bodies didn’t fall at the same rate under the same gravitational force, then a free-falling observer could not “unmake” gravity, and the whole conception of general relativity would fall apart.

爱因斯坦几乎立刻就看出了引力和加速度之间的等价性——他称之为“等效原理”——意味着引力必定会使光线弯曲。牛顿也曾提出过这种观点,因为他认为光是由受引力影响的物质粒子组成的,因此它们会像伽利略和哈里奥特的炮弹一样遵循抛物线轨迹。但爱因斯坦在 1905 年就已证明光粒子(光子)没有静止质量,因此他的论点与牛顿的论点截然不同。你可以在图 12.1中看到它的核心,它说明了“等效原理”。此外,由于光在弯曲路径上传播时比在直线路径上传播时需要更长的时间才能到达远处的观察者,因此重力似乎会减慢光速。这与狭义相对论非常不同,狭义相对论认为光速是恒定的,对所有观察者来说都是相同的。(在局部惯性系(例如图 12.1 (a))中测量的局部真空光速仍然是常数c ,但观察者测量到的光从远处光源沿弯曲路径传播时的速度较慢。)

Almost immediately Einstein saw that this equivalence between gravity and acceleration—what he called “the principle of equivalence”—meant that gravity must bend light rays. Newton had suggested this, too, because he believed light was made of material particles that would be affected by gravity, so they’d follow a parabolic trajectory like Galileo’s and Harriot’s cannonballs. But Einstein had shown in 1905 that light particles—photons—have no rest mass, so his argument was very different from Newton’s. You can see the nub of it in figure 12.1, which illustrates the “principle of equivalence.” And there’s more: since light takes longer to reach a distant observer when it travels on a curved rather than a straight path, gravity appears to slow down the speed of light. This is very different from special relativity, where the speed of light is constant, the same for all observers. (The local vacuum light speed, measured in a locally inertial frame such as figure 12.1(a), is still the constant c, but the observer measures a slower speed for light traveling on the curved path from a distant source.)

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图 12.1。等效原理。该图显示了三个参考系:(a)“惯性”系,例如无重力(自由漂浮)的宇宙飞船或自由落体的人或电梯,(b)以 32 英尺/秒/秒(或 9.8 米/秒/秒)的速度向上加速的宇宙飞船或电梯,以及(c)地球上的房间。在(b)中,当宇宙飞船或电梯加速向上时,其地板会向上推你,你会感觉和在地球上一样重(框架 (c)),这会对我们施加向上的推力,以响应我们向下的重力加速度 32 英尺/秒/秒或 9.8 米/秒/秒。(这些值是海平面地球表面的重力加速度——当然,在更高的海拔高度,加速度会根据牛顿的平方反比定律减小。)

FIGURE 12.1. The principle of equivalence. The diagram shows three frames of reference: (a) an “inertial” frame, such as a gravity-free (free-floating) spaceship or a free-falling person or elevator, (b) a spaceship or elevator accelerating upward at 32ft/sec/sec (or 9.8m/sec/sec), and (c) a room here on Earth. In (b), with the floor of the spaceship or elevator pushing up on you as it accelerates upward, you would feel just as heavy as you do on Earth (frame (c)), which pushes up on us in response to our downward gravitational acceleration of 32ft/sec/sec or 9.8m/sec/sec. (These values are for the gravitational acceleration on the surface of Earth at sea level—at higher altitudes, of course, the acceleration decreases according to Newton’s inverse square law.)

在惯性系 (a) 中,若没有力的作用,抛射物一旦发射就会以恒定速度沿直线运动。这是牛顿第一定律,即惯性定律,因此有“惯性系”一说。在 (c) 中,抛射物的路径向下,正如哈里奥特和伽利略所展示的,在 (b) 中也有同样的情况,与 (c) 等价。(当宇宙飞船加速向上时,地板会升起与抛射物相遇,因此外部观察者会看到抛射物靠近地板。换句话说,它的路径看起来是弯曲的。)同样,从火箭或电梯的一侧发送到另一侧的光线在 (a) 系中是直线的,但在 (b) 中,地板会升起与光线相遇,因此在外部观察者看来光线是弯曲的。由于 (b) 等同于 (c),爱因斯坦推断重力会使光线弯曲。

In the inertial frame (a), in the absence of forces, once launched a projectile will move in a straight line with constant speed. This is Newton’s first law—the law of inertia, hence the term “inertial frame.” In (c), a projectile has a downward parabolic path, as Harriot and Galileo showed—and the same happens in (b), which is equivalent to (c). (As the spaceship accelerates upward, the floor comes up to meet the projectile, so an outside observer sees the projectile move closer to the floor. In other words, its path appears bent.) Similarly, a light ray sent from one side of the rocket or elevator to the other will be straight in frame (a), but the floor in (b) will rise to meet it so that it appears bent to an outside observer. Since (b) is equivalent to (c), Einstein deduced that gravity bends light.

早在 1911 年,当时还是布拉格德国大学教授的爱因斯坦6就发表了一个数学推导公式,量化了光在太阳附近的弯曲程度。在这篇论文中,他还发展了 1907 年提出的另一个想法:引力红移,即引力导致光频率向光谱较长波长(或红色)端移动。它类似于多普勒效应,这种效应使救护车或警车上的警报器具有独特的声音:车辆加速驶近时声音较高,驶离时声音较低。这是因为声波首先相对于您被压缩,使其波长变短,然后随着车辆后退而扩展——光波也是如此。想象一下图 12.1(b)中的航天器底部向顶部的观察者发送的光脉冲。当光向上传播时,观察者与航天器一起加速向上,因此他们正在远离光发射点;这意味着光的波长看起来被拉长或变红了。但是由于图 12.1(b)(c)是“等效的”,所以 (c) 中梯子顶端的观察者应该看到来自地面光源的红移光,因为地面上的重力更强。这是引力红移存在的一个简单论据!(然而,由于等效原理只在局部成立,因此计算结果只是近似的。爱因斯坦使用的另一个红移论据是,光在逆着引力场传播时会损失能量。但他需要他的最终理论来进行精确的预测计算。)

As early as 1911, when he was a professor at the German University in Prague,6 Einstein published a mathematical derivation quantifying the amount of light bending near the Sun. In this paper he also developed another idea he’d had back in 1907: the gravitational redshift, where gravity causes a shift in light frequency toward the longer-wavelength (or red) end of the spectrum. It’s analogous to the Doppler effect that gives the sound of a wailing siren on an ambulance or police car its distinctive sound: higher as the vehicle accelerates toward you, lower as it moves away. That’s because the sound waves are first compressed relative to you, making their wavelength shorter, and then expanded as the vehicle recedes—and the same happens with light waves. Imagine a pulse of light sent from the bottom of the spacecraft in figure 12.1(b) to an observer at the top. As the light travels upward, the observer is accelerating upward with the spacecraft, so they are moving away from the point where the light was emitted; this means the light’s wavelength appears elongated or reddened. But since figures 12.1(b) and (c) are “equivalent,” an observer at the top of a ladder in (c) should see redshifted light from a source on the ground, where gravity is stronger. This is a simple argument for the existence of gravitational redshift! (The resulting calculations are only approximate, though, since the equivalence principle holds only locally. Another redshift argument, which Einstein used, is that light loses energy traveling against a gravitational field. But he would need his final theory for exact, predictive calculations.)

如果引力相对论地使光红移到更长的波长,那么它也必须对其他振动产生同样的效果——包括时钟的滴答声。这意味着在较强的引力场中,远距离观察者看到的时钟走得更慢(因为滴答声之间的时间间隔更长)。现在,这一事实——以及狭义相对论的时间膨胀(由洛伦兹变换提供)——在校准 GPS 方向时被考虑在内。然而,在 1911 年,爱因斯坦的重点是以后技术成熟时可能被检验的预测:事实上,他提出了红移论证来检验时间变慢的理论,而他早在 1907 年就已直觉地意识到了这一点。

If gravity relativistically redshifts light to a longer wavelength, it must do the same for other vibrations, too—including the ticking of a clock. Which means that distant observers see clocks run more slowly in a stronger gravitational field (because there is a longer time between ticks). This fact is now taken into account—along with special relativity’s time dilation, courtesy of the Lorentz transformations—in calibrating GPS directions. In 1911, though, Einstein’s focus was on predictions that might be tested later, when the technology was up to it: in fact, he’d come up with the redshift argument to test the slowing of time idea, which he’d already intuited in 1907.

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1911 年,爱因斯坦开始更深入地探索他的思想。他需要能够将他的直觉思维实验转化为可测试的方程,而且由于引力可以减慢光的视速,他想知道这种可变的光速是否可能代表他新理论中的“引力势”。我在第 6 章中提到,约瑟夫-路易斯·拉格朗日已经用标量“势” V的形式制定了牛顿引力的平方反比定律。它很容易从牛顿定律开始,正如你在本尾注中看到的那样,7或从高斯的通量定律(类似于麦克斯韦在第 6 章中推导的 ∇ ∙ E = 4πρ )得出。无论哪种方式,牛顿引力定律都可以表示为

In 1911 Einstein began to explore his ideas in more depth. He needed to be able to transfer his intuitive thought experiments into testable equations, and since gravity can slow down the apparent speed of light, he wondered if this variable light-speed might represent the “gravitational potential” in his new theory. I mentioned in chapter 6 that Joseph-Louis Lagrange had formulated Newton’s inverse-square law of gravity in terms of a scalar “potential,” V. It comes out easily starting from Newton’s laws, as you can see in this endnote,7 or from Gauss’s flux law (analogously to Maxwell’s derivation of ∇ ∙ E = 4πρ in chap. 6). Either way, Newton’s law of gravity can be expressed as

22+22+222=4πρ

2Vx2+2Vy2+2Vz22V=4πGρ

这里 ρ 是产生引力场的物质的密度。它类似于麦克斯韦方程中的电荷密度,但有相同的注意事项。8 G是牛顿 平方反比定律中的比例引力常数。

Here ρ is the density of the matter producing the gravitational field. It’s analogous to the charge density in Maxwell’s equations, with the same caveats.8 G is the gravitational constant of proportionality in Newton’s inverse square law.

这种表述的最初优势在于,它把势能视为一个连续的标量“场”(用法拉第-麦克斯韦的语言来说),而不是平方反比定律中隐含的超距作用观点,后者考虑了引力粒子之间的数值距离,但没有考虑它们之间整个空间的性质。事实上,由于平方反比定律给出了给定质量对放置在周围空间中任意一点的单位测试质量施加的力,因此它可以可以解释为定义矢量场;不过,使用标量势进行计算通常更简单。因此,爱因斯坦想知道他的可变光速是否可以在他的新理论中发挥与势V在牛顿理论中发挥的作用类似的作用——然后他继续创建了一个“静态”引力理论。“静态”意味着不随时间变化,因此它描述了一颗静止的恒星,或者地球上弱的、本质上均匀的引力场。但爱因斯坦仍在摸索,他(错误地)假设在静态引力场中,时空的空间部分将是欧几里得的,就像地球上的日常计算一样。这意味着虽然闵可夫斯基度量描述了狭义相对论中的时空,但爱因斯坦静态引力场中描述时空的相关度量将“几乎是闵可夫斯基的”:

The original advantage of this formulation was that it treats the potential as a continuous scalar “field,” to use Faraday-Maxwell language, rather than the action-at-a-distance view that seemed implicit in the inversesquare law, which takes into account the numerical distance between the gravitating particles but not the nature of the whole space between them. In fact, since the inverse square law gives the force exerted by a given mass on a unit test mass placed at any point in the surrounding space, it can be interpreted as defining a vector field; still, the calculations are often simpler using the scalar potential. So, Einstein wondered if his variable speed of light could play a similar role in his new theory as the potential V plays in Newton’s—and he went on to create a “static” gravitational theory. “Static” means unchanging in time, so it describes a quiescent star, say, or the weak, essentially uniform gravitational field here on Earth. But Einstein was still feeling his way, and he assumed (wrongly) that in a static gravitational field, the spatial part of space-time would be Euclidean, as it is for everyday calculations on Earth. This meant that while the Minkowski metric describes the space-time in special relativity, the relevant metric describing space-time in Einstein’s static gravitational field would be “almost Minkowskian”:

ds 2 = dx 2 + dy 2 + dz 2 − ( c ( x, y, z )) 2 dt 2 ,

ds2 = dx2 + dy2 + dz2 − (c(x, y, z))2dt2,

其中,光速c现在是空间坐标的函数。这完全是个错误,他后来才意识到这一点。与此同时,他在 1911 年发表了这一静态理论。

where c, the speed of light, is now a function of the spatial coordinates. This is all a mistake, as he’ll realise later. Meantime, he published this static theory in 1911.

总之,在布拉格待了一年多一点的时间里,他发表了六篇关于相对论的论文(以及五篇其他论文)。他把想法公之于众,征求有助于改进这些想法的反馈意见——尽管他从尖嘴利舌的马克斯·亚伯拉罕那里得到的反馈比他预想的要多一点。亚伯拉罕是电磁学专家——他出版了德国关于麦克斯韦理论的领先教科书,还创建了新发现的电子的开创性(但现已过时)模型。但他坚信亨德里克·洛伦兹和亨利·庞加莱的半相对论以太电磁学方法,当爱因斯坦的狭义相对论抛弃了这种方法时,他非常沮丧。现在,在 1911 年,他用他敏锐的头脑和不幸的锋利的笔,公开批评爱因斯坦推广相对论的尝试。然而,他不仅仅是一个吹毛求疵的批评家,令人惊讶的是,他在 1912 年初发表了自己的相对论引力理论。

All up, in the little more than a year that he spent in Prague, he published six papers on relativity (and five other papers). He was putting ideas out there, inviting feedback that would help refine them—although he got a little more than he bargained for from the sharp-tongued Max Abraham. Abraham was an expert on electromagnetism—he had published Germany’s leading textbook on Maxwell’s theory, and he’d also created a pioneering (but now obsolete) model of the newly discovered electron. But he was a firm believer in Hendrik Lorentz and Henri Poincaré’s semirelativistic ether-based approach to electromagnetism, and he’d been mightily upset when Einstein’s special relativity dispensed with it. Now, in 1911, he applied his sharp mind, and his unfortunately sharp pen, to public criticisms of Einstein’s attempts to generalise relativity. Yet he was no mere carping critic, and surprisingly, in early 1912 he published his own relativistic theory of gravity.

虽然爱因斯坦静态理论中的度量“几乎”是闵可夫斯基度量,但亚伯拉罕的理论完全是闵可夫斯基的——而且最初爱因斯坦被它迷住了。(之前已经有其他人尝试将引力纳入狭义相对论,包括庞加莱和闵可夫斯基。)正如爱因斯坦告诉他的老朋友、瑞士理工学院的毕业生、他长期的洞察力顾问米歇尔·贝索,“一开始(14 天)我也完全被 [亚伯拉罕] 公式的优美和简单所迷惑。”但随后爱因斯坦发现了“推理中的一些严重错误……这就是当一个人不进行物理思考而进行形式化 [数学] 运算时会发生的情况!” 9

While the metric in Einstein’s static theory was “almost” the Minkowski metric, Abraham’s theory was fully Minkowskian—and initially Einstein was captivated by it. (There had already been other attempts to fit gravity into special relativity, including by Poincaré and Minkowski.) As Einstein told his old friend Michele Besso, a fellow graduate from the Swiss Polytechnic and his longtime insightful sounding-board, “At first (for 14 days) I too was completely bluffed by the beauty and simplicity of [Abraham’s] formulas.” But then Einstein spotted “some serious mistakes in reasoning.... This is what happens when one operates formally [mathematically] without thinking physically!”9

几个月来,他和亚伯拉罕在《物理学年鉴》上争论不休,这可谓 20 年前《自然》杂志上矢量之争的微型版。亚伯拉罕的尖刻言辞使他与大多数物理学家疏远,但直言不讳的爱因斯坦同情亚伯拉罕的口无遮拦的性格,而他的批评也确实帮助爱因斯坦的思想更加清晰。1912 年 7 月,就在爱因斯坦离开布拉格前往苏黎世和联邦理工学院之前,他发表了一篇论文,介绍他尝试应用等效原理寻找新引力方程的经历,并补充说:“我想让我的所有同事都尝试一下这个重要问题!”但亚伯拉罕冷嘲热讽地回应道:“爱因斯坦为明天的相对论祈求功劳,并呼吁同事们保证这一点。”此时,爱因斯坦放弃了辩论,但并没有放弃自己为新的相对论引力理论而进行的斗争,也必须说,他对亚伯拉罕作为物理学家的能力的尊重。10

For months he and Abraham traded arguments in the pages of Annalen der Physik, in a mini version of the vector wars in Nature two decades earlier. Abraham’s caustic tongue had alienated him from most of the physics community, but the outspoken Einstein had sympathy for Abraham’s tendency to run off at the mouth, and his criticisms did help sharpen Einstein’s ideas. In July 1912, just before he left Prague for Zurich and the ETH, Einstein published a paper on his attempts to apply the equivalence principle in his search for a new equation of gravity, adding, “I would like to ask all of my colleagues to have a try at this important problem!” But Abraham responded snidely, “Einstein begs credit for the theory of relativity of tomorrow and appeals to his colleagues so that they may guarantee it.” At which point Einstein gave up the debate, but not his own battle for a new relativistic theory of gravity—nor, it must be said, his respect for Abraham’s ability as a physicist.10

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这基本上就是爱因斯坦 1912 年 8 月抵达苏黎世时在广义相对论方面的进展情况。他直观地推导出等效原理及其对时间和光速测量的影响,这完全是灵感的源泉——但绝妙的想法是一回事,证明它们又是另一回事。例如,引力红移的直接实验证明直到 1960 年才出现,尽管爱因斯坦没有亲眼看到它:他于 1955 年去世。但在 1912 年,证明只是一个小问题,而爱因斯坦为寻找完全相对论引力理论的方程式所付出的努力则微不足道。他的天才在于物理思维,而不是数学,尽管他实际上是一个非常优秀的数学家,即使不是一个知识渊博的人——他逃了太多的闵可夫斯基的课。所以,不久之后,他就向他的老同学恳求道:“格罗斯曼,你必须帮助我,否则我会发疯的!” 11

This was, essentially, the state of Einstein’s progress toward general relativity when he arrived in Zurich in August 1912. His intuitive deductions of the equivalence principle and its consequences for the measurement of time and light speed were utterly inspired—but brilliant ideas were one thing; proving them was quite another. Direct experimental proof of the gravitational redshift, for example, wouldn’t come until 1960, although Einstein didn’t live to see it: he died in 1955. But in 1912 proof was a minor problem compared with Einstein’s struggle to find the equations for a fully relativistic theory of gravity. His genius was in physical thinking, not maths, although he was actually a very good mathematician, if not a wellinformed one—he’d skipped too many of Minkowski’s classes. So, it wasn’t long before he pleaded with his old classmate, “Grossmann, you must help me, otherwise I’ll go crazy!”11

图像

阿尔伯特·爱因斯坦,1912 年。苏黎世联邦理工学院图书馆,Bildarchiv/摄影师:Jan. F. Langhans/ Portr_05936。公共领域。

Albert Einstein in 1912. ETH-Bibliothek Zürich, Bildarchiv/Photographer: Jan. F. Langhans/ Portr_05936. Public domain.

爱因斯坦为何需要帮助

WHY EINSTEIN NEEDED HELP

让爱因斯坦抓狂的问题是:如何调和等效原理(图 12.1(b)(c)中的观察者无法分辨他们是在平稳加速的电梯中还是在引力场中的封闭房间中)与相对论原理(物理定律对所有观察者来说应该具有相同的形式)。12的静态引力理论是等效原理的结果,他从中推导出了充当引力势能角色的光速可变性。但这似乎与相对论原理相矛盾,因为显然它与狭义相对论不一致,我们在图 9.3中看到的洛伦兹变换依赖于光速恒定。(爱因斯坦的新广义相对论必须与狭义相对论相吻合,以涵盖相对加速度为零的情况。)

The problem driving Einstein crazy was this: how to reconcile the principle of equivalence—where the observers in figures 12.1(b) and (c) can’t tell if they’re in a smoothly accelerating elevator or a closed room in a gravitational field—with the principle of relativity, where the laws of physics should have the same form for all observers.12 His static theory of gravity was a consequence of the principle of equivalence, from which he’d deduced the variable speed of light that played the role of the gravitational potential. But this seemed to contradict the principle of relativity, because clearly it didn’t gel with the special theory, where the Lorentz transformations we met in figure 9.3 rely on the constant speed of light. (Einstein’s new general theory of relativity would have to fit with the special theory, to cover cases where the relative acceleration is zero.)

而且,由于狭义相对论中任意两个观察者的相对速度都是恒定的,所以牛顿第一定律(惯性定律)对他们俩都成立。正如我们在第三章中看到的那样,牛顿用物体速度的变化来定义力,因此惯性定律说,在没有力的情况下,静止的物体将保持静止,而以恒定速度运动的物体将保持以恒定速度运动。将这个定律应用到狭义相对论中,如果一个物体在观察者A的参考系中处于静止状态,那么根据观察者B 的说法,它处于恒定运动中,反之亦然;无论哪种情况,惯性定律对两个观察者都成立,因此就像在牛顿物理学中一样,他们的参考系是“惯性的”。然而,当相对运动加速时,情况就不再如此了。

What’s more, because the relative speed of any two observers is constant in special relativity, Newton’s first law (of inertia) holds for both of them. As we saw in chapter 3, Newton defined force in terms of the change in a body’s velocity, so the law of inertia says that in the absence of forces, an object at rest remains at rest, and an object moving with constant velocity remains moving with that constant velocity. Applying this in special relativity, if an object is at rest in the frame of observer A, then it is in constant motion according to observer B, and vice versa; either way, the law of inertia holds for both observers, and so just as in Newtonian physics, their frames of reference are “inertial.” When the relative motion is accelerated, however, this is no longer the case.

这表明爱因斯坦认为,没有经验理由将广义相对论与狭义相对论的惯性参考系联系起来。13意味着不再存在一个单一的、有限的坐标变换群,使物理定律保持不变,就像狭义相对论中的洛伦兹变换一样。相反,自然定律应该在任何平滑的坐标变换下保持不变。14由于没有特殊的坐标变换群,因此也没有保持不变的特殊度量。因此,必须用一个完全通用的度量来取代闵可夫斯基度量,

This suggested to Einstein that there was no empirical reason to tie a general theory of relativity to the inertial reference frames of the special theory.13 Which meant there was no longer a single, finite group of coordinate transformations that kept the laws of physics invariant, the way the Lorentz transformations do in special relativity. Rather, the laws of nature should be invariant under any smooth coordinate transformation.14 Since there was no special group of coordinate transformations, there was no special metric that remained invariant. So, the Minkowski metric had to be replaced by a completely general one,

ds 2 = g μν dx μ dx ν

ds2 = gμνdxμdxν,

其中系数g μν通常是坐标的函数。(因此,闵可夫斯基度量只是此一般度量的一个特例。)

where the coefficients gμν are, in general, functions of the coordinates. (So, the Minkowski metric is just one special case of this general metric.)

这意味着,用这个度量描述的时空一般来说必须是弯曲的——尽管爱因斯坦直到格罗斯曼向他介绍了黎曼几何和我们在第 10 章中遇到的曲率条件(即由g μν的导数构建的黎曼张量不为零)后才意识到这一点。在布拉格,爱因斯坦已经想出了从基于长度收缩的直观论证来看,时空必须是非欧几里得的,但他一直苦恼的事情之一是,在完全通用的度量中,x μ可以是任何类型的坐标。这意味着dx μ不一定与时间和空间测量直接相关,就像它们在欧几里得和闵可夫斯基度量的标准(笛卡尔)形式中一样,其中dx μ是时间和空间差异dt、dx、dy、dz

This meant that the space-time described by this metric must, in general, be curved—although Einstein only realised this when Grossmann introduced him to Riemannian geometry and the condition for curvature we met in chapter 10 (namely, that the Riemann tensor, which is built from derivatives of the gμν, is not zero). In Prague, Einstein had already figured out that space-time must be non-Euclidean, from an intuitive argument based on length contraction, but one of the things he’d been tearing his hair over was that in a completely general metric, the xμ could be any kind of coordinates. Which meant that the dxμ didn’t necessarily relate directly to time and space measurements, as they do in the standard (Cartesian) form of the Euclidean and Minkowski metrics, where the dxμ are the time and space differences dt, dx, dy, dz.

爱因斯坦“对此深感困扰”,他后来回忆道15 — —直到他返回苏黎世之前。我在第 9 章中指出,在狭义相对论中,尽管笛卡尔坐标确实与闵可夫斯基时空中的时间和空间有关,但相对移动的观察者之间对实际时间和距离测量并不一致(如图9.3 方框中的洛伦兹变换所示)。相反,我们看到表达式x 2 + y 2 + z 2 − ( ct ) 2 — —以及闵可夫斯基度量ds 2 = dx 2 + dy 2 + dz 2 − ( cdt ) 2 — —是不变的。换句话说,两个观察者只对ds达成一致,即度量给出的时空中事件之间的间隔(或“距离”)。爱因斯坦没有在他的狭义相对论中使用区间ds — — 该区间来自闵可夫斯基和高斯,正如我们在第 10 章中看到的,爱因斯坦花了一段时间才接受这种几何“度量”方法。(在他 1905 年的论文中,他专注于洛伦兹变换的代数后果。)但就在离开布拉格之前,他意识到了ds和等效原理的一些根本重要的东西。

Einstein was “much bothered by this,” as he later recalled15—until just before he returned to Zurich. I showed in chapter 9 that in special relativity, even though Cartesian coordinates do relate to time and space in Minkowski space-time, there is no agreement between relatively moving observers on actual time and distance measurements (as the Lorentz transformations in the box for fig. 9.3 show). Rather, we saw that it is the expression x2 + y2 + z2 − (ct)2—and, therefore, the Minkowski metric, ds2 = dx2 + dy2 + dz2 − (cdt)2—that is invariant. In other words, the two observers only agreed on ds, the interval (or “distance”) between events in space-time, as given by the metric. Einstein hadn’t used the interval ds in his special theory—that had come from Minkowski and Gauss, as we saw in chapter 10, and Einstein had taken a while to warm to this geometric “metric” approach. (In his 1905 paper he’d focussed on algebraic consequences of the Lorentz transformations.) But just before leaving Prague, he’d realised something fundamentally important about ds and the principle of equivalence.

在引力场或加速参考系中,当没有其他力作用时,物体和光子必须沿弯曲的测地线运动,而不是沿惯性系中的直线运动。这最终是爱因斯坦严格表达图 12.1中所示的等效原理的方式。因为,正如我们在第 10 章中看到的,两点之间的最短距离位于测地线上,这可以通过最小化(或者说极化)16 “距离” s = ∫ds来找到。爱因斯坦突然意识到,这意味着在他的理论中具有物理意义的是ds整个度规,而不是度规中单独的坐标微分。如果它要具有物理意义,则所有观察者必须对此达成一致,就像他们对狭义相对论中的闵可夫斯基度量达成一致一样。但是,当爱因斯坦只有度量的一般形式时,他如何表达这种不变性呢?他感到困惑。

In a gravitational field or accelerating reference frame, when no other forces are acting, material objects and light photons must travel on curved geodesics, rather than the straight lines they take in inertial frames. This was, finally, Einstein’s way of rigorously expressing the principle of equivalence sketched in figure 12.1. For, as we saw in chapter 10, the shortest distance between two points lies on a geodesic, and this is found by minimising (or, rather, extremising)16 the “distance” s = ∫ds. As Einstein suddenly realised, this means that it is ds, or the metric as a whole, which has physical meaning in his new theory—not the separate coordinate differentials in the metric. And if it is to have physical meaning, all observers must agree on it, as they do with the Minkowski metric in special relativity. But how could Einstein express this invariance when he only had a general form of the metric? He was flummoxed.

于是,忠诚的格罗斯曼去了图书馆,想看看是否有人找到了保持一般度量和方程不变的方法。就这样,他和爱因斯坦读到了里奇和列维-奇维塔关于张量分析的里程碑式论文。

So, loyal Grossmann went off to the library, to see if anyone had figured out a way to keep general metrics and equations invariant. And that’s how he and Einstein came to read Ricci and Levi-Civita’s landmark paper on tensor analysis.

爱因斯坦和格罗斯曼开始研究张量微积分

EINSTEIN AND GROSSMANN TAKE UP TENSOR CALCULUS

格罗斯曼对数学充满热情,但对物理并不怎么感兴趣,直到他被爱因斯坦的设想“迷住”。17即使在早期阶段,这也是一个多么了不起的设想。我们已经看到他根据等效原理做出的非凡推论,但爱因斯坦还倾向于引力和几何密不可分的激进观点。这是因为度量必须同时描述时空的弯曲几何形状引力场本身——因为当没有引力时,你就会回到平坦时空的闵可夫斯基度量,物体会在直线而不是弯曲的测地线上移动。所以,引力影响时空的几何形状——曲率,而曲率描述引力。至少,如果爱因斯坦能找到正确的方程的话,情况会是这样。

Grossmann was passionate about mathematics, but he hadn’t cared much for physics until he “caught fire” with Einstein’s vision.17 And what a remarkable vision it was, even at this early stage. We’ve already seen the extraordinary deductions he made from the principle of equivalence, but Einstein was also homing in on the radical idea that gravity and geometry were inextricably entwined. That’s because the metric had to describe both the curved geometry of space-time and the gravitational field itself—for when there’s no gravity, you’re back to the Minkowski metric of flat spacetime and to objects moving on straight lines instead of curved geodesics. So, gravity affected the geometry—the curvature—of space-time, and the curvature described the gravity. At least, it would if Einstein could ever find the right equations.

再次,他的出发点是牛顿引力方程,以引力势来表示:∇ 2 V = 4π G ρ。毕竟,任何新的引力理论都必须符合牛顿在地球和太阳系其他远离强引力源的弱场中取得巨大成功的理论。爱因斯坦放弃了他早先认为光速就是引力势的想法,因为它与狭义相对论不一致,但现在他做出了一个大胆的断言:

Once again, his starting point was Newton’s equation of gravity, expressed in terms of a gravitational potential: ∇2V = 4πGρ. After all, any new theory of gravity had to fit with Newton’s spectacularly successful theory in the weak, fairly uniform fields here on Earth and elsewhere in the solar system far away from strong sources of gravity. Einstein had abandoned his earlier idea that the speed of light was the gravitational potential because it didn’t gel with special relativity, but now he made a daring assertion: the

度量系数g μν 本身不仅代表时空的度量特性,还代表牛顿引力势V的相对论类似物。(这就是为什么他和格罗斯曼使用字母g来表示弯曲时空的度量系数,而我已经将其抄袭过来了。)

metric coefficients gμν themselves represent not only the metric properties of space-time but also the relativistic analogue of the Newtonian gravitational potential V. (That’s why he and Grossmann used the letter g that I’ve already cribbed for the metric coefficients of curved space-time.)

从概念上讲,这是大胆的——因为,正如他后来意识到的,它明确了几何与引力之间的联系——从数学上讲,这也是大胆的,因为它表明了数学物理中前所未有的复杂性。部分原因是在 4 维时空中,每个指标都取 1 到 4 的值,双指标张量有 4 × 4 = 16 个不同的分量,尽管对于度量张量,指标应该是对称的,g μν = g νμ (因为标量积是可交换的,正如我们在第 11 章末尾看到的那样)——这意味着只有十个独立的度量系数:

It was daring conceptually—because, as he realised later, it made explicit the link between geometry and gravity—and it was daring mathematically, because it suggested a complexity never seen before in mathematical physics. That’s partly because in 4-D space-time, where the indices each take on the values from 1 to 4, a two-index tensor has 4 × 4 = 16 different components, although for the metric tensor, the indices should be symmetric, gμν = gνμ (because the scalar product is commutative, as we saw near the end of chap. 11)—which means there are only ten independent metric coefficients:

g 11g 12g 13g 14g 22g 23g 24g 33g 34g 44

g11, g12, g13, g14, g22, g23, g24, g33, g34, g44.

但那仍然是十个引力势,因此必须有十个引力场方程,而不是单个牛顿方程 ∇ 2 V = 4π G ρ。多么大的挑战啊!从哪里开始呢?

But that’s still ten gravitational potentials, so there’d have to be ten gravitational field equations instead of the single Newtonian equation ∇2V = 4πGρ. What a challenge! And where to begin?

首先,在牛顿方程中,等式左边的引力势与 ρ 有关,ρ 是引力的物质密度,而物质的密度就是等式右边的引力源。对于目前正在密切合作解决这一问题的爱因斯坦和格罗斯曼来说,左边表达式的明显类比似乎是某种由势 g μν的二阶导数构成的十分量张量表达式,就像 ∇ 2 V中的二阶导数一样。

Well, for a start, in the Newtonian equation the gravitational potential on the left-hand side of the equation relates to ρ, the density of the matter that is the source of the gravity represented on the right. For Einstein and Grossmann, who were now collaborating closely on the problem, the obvious analogy for the left-hand expression seemed to be some sort of ten-component tensor expression made from the second derivatives of the potentials gμν, like the second derivatives in ∇2V.

不久之后,格罗斯曼发现“里奇张量” R μν就是这样一个庞然大物。里奇在 1903 年发现了这个张量——尽管他没有给它命名;通常是后来的研究人员赋予了这种尊贵的数学和科学术语。里奇张量是我在第 10 章中提到的四指标黎曼张量的收缩,我们在第 11 章中看到,“收缩”意味着你将上指标和下指标设置为相等并对它们求和:RμRμαα。这看起来没什么大不了的,不是吗?事实上,由于黎曼张量本身就是度量系数导数的总和,所以R μν是数百项的总和!其中许多项相互抵消或为零,但尽管如此,广义相对论中仍然有大量的计算;到 20 世纪 80 年代末,计算机代数软件包的日益普及和复杂化成为研究人员的福音,这一点你可以想象得到。

It wasn’t long before Grossmann found that the “Ricci tensor,” Rμν, was such a beast. Ricci had discovered this tensor in 1903—although he didn’t name it; it is usually later researchers that bestow such honorific mathematical and scientific terminology. The Ricci tensor is the contraction of the fourindex Riemann tensor I mentioned in chapter 10, and we saw in chapter 11 that “contraction” means that you set an upper and lower index equal and sum over them: RμvRμαvα. It looks innocuous, doesn’t it? In fact, since the Riemann tensor is itself the sum of derivatives of the metric coefficients, Rμν is the sum of hundreds of terms! Many of them cancel or are zero, but nonetheless there’s an awful lot of calculating in general relativity; by the late 1980s the increasing availability and sophistication of computer algebra packages were becoming a boon to researchers, as you can imagine.

顺便说一句,我们很快就会看到,与矢量方程不同,张量方程最好用一般分量来表示,因为张量指标对于计算至关重要——例如给出里奇张量的收缩。所以从现在开始,我通常会遵循这样写张量的常见做法:g μν表示度量张量,R μν表示里奇张量,Rμαα对于黎曼张量等等,而不是使用整个张量标签(例如g、Ric、Riem)的更正确表示。因此,当您看到使用一般指标(例如 μ、ν、...)的方程时,您知道它对所有指标值都成立。您还知道它在坐标变换下是形式不变的(协变的),而以特定分量表示的方程(指标用数字指定,如R 12,或用特定坐标指标指定,如速度的x分量的v x或磁场的 y 分量)不是

By the way, we’ll see soon that unlike vector equations, tensor equations are best written in terms of general components, because tensor indices are so vital for computations—such as the contraction that gives the Ricci tensor. So from now on I’ll generally follow the common practice of writing tensors this way: gμν for the metric tensor, Rμν for the Ricci tensor, Rμαvα for the Riemann tensor, and so on—rather than their more correct representation using whole-tensor labels such as g, Ric, Riem, say. So, when you see an equation using general indices such as μ, ν, … , you know it holds for all values of the indices. You also know it is form-invariant (covariant) under coordinate transformations, whereas equations in terms of specific components—with indices designated by numbers as in R12, or by specific coordinate indices as in vx for the x-component of velocity, say, or By for the y-component of the magnetic field—are not.

• • •

• • •

爱因斯坦和格罗斯曼将里奇张量作为方程左边的候选,而右边则需要一个与牛顿质量密度 ρ 类似的张量。质量密度是描述产生引力场的物质分布的一种方式,但显然能量也应该是其中的一部分,这是E = mc 2的结果。至于如何将其转化为张量,这两位合作者的前数学教授闵可夫斯基在以张量形式写出麦克斯韦方程时就已经指明了方向。

With the Ricci tensor as a candidate for the left-hand side of his equation, Einstein and Grossmann needed an analogue of the Newtonian mass density ρ for the right-hand side. The mass density is a way of characterising the distribution of the matter producing a gravitational field, but obviously energy should be part of the mix, too, courtesy of E = mc2. As to how to put this into a tensor, the two collaborators’ former maths professor, Minkowski, had, in fact, already pointed the way, when he wrote Maxwell’s equations in tensor form.

在通常的矢量形式的麦克斯韦方程中,电场矢量E和磁场矢量B是交织在一起的:在旋度方程中,方程的一边是E ,另一边是B。闵可夫斯基所做的就是将这些交织在一起的三维场组合成平坦四维时空中的单个张量。正如我在第 9 章中提到的,张量当时还没有成为主流——直到爱因斯坦在 1916 年发表广义相对论基础概述之后,“张量”这个名称才在这个语境中被广泛使用。因此,1910 年,当索末菲效仿闵可夫斯基,将电场矢量EB组合成一个量F (通常用其广义坐标F μν表示)时,他使用了术语“六矢量”而不是“张量”。这是因为F μν有六个独立分量,因为这只是EB中信息的另一种写法,它们各自有三个分量,分别代表x、y、z轴。令人惊讶的是,即使在 1916 年的概述中,爱因斯坦也将F μν称为“六矢量”,尽管他在其他地方使用了“张量”。但他确实使用了我在这里使用的现在常见的符号——除了他使用 ∂ 而不是现代的偏导数逗号,我们在第 11 章末尾看到过。我将列出这六个电磁张量分量,以便您了解这个想法,但只需将它们视为定义:

In the usual vectorial form of Maxwell’s equations, the electric and magnetic field vectors E and B are intertwined: in the curl equations you have E on one side of the equation and B on the other. What Minkowski did was to combine these intertwined 3-D fields into a single tensor in flat 4-D space-time. As I mentioned in chapter 9, tensors were not yet in the mainstream—and the name “tensor” wasn’t generally used in this context until after Einstein published his 1916 overview of the foundations of general relativity. So, in 1910, when Sommerfeld followed Minkowski’s lead and combined the electric and magnetic field vectors E and B into a single quantity, F (usually denoted by its general coordinate Fμν), he used the term “six-vector” rather than “tensor.” That’s because Fμν has six independent components, since it is just another way of writing the information in E and B, which each have three components, one for each of the axes x, y, z. Surprisingly, even in his 1916 overview Einstein, too, called Fμν a “six-vector,” although he used “tensor” elsewhere. But he did use the now common notation I’m using here—aside from the fact that he used ∂ rather than the modern commas for partial derivatives, which we saw at the end of chapter 11. I’ll list these six electromagnetic tensor components so you can see the idea, but just take them as definitions:

F 14 = −E x,F 24 = −E y F 34 = −E z F 12 = B z,F 31 = B y,F 23 = B x

F14 = −Ex, F 24 = −E y, F 34 = −Ez, F12 = Bz, F 31 = B y, F 23 = Bx;

另外,F μν = - F νμ,且F μμ = 0。

in addition, Fμν = −F νμ, and Fμμ = 0.

有了这些定义,第 8 章中四个漂亮的麦克斯韦方程变得更加简洁,更加优雅。我在这里给出它们只是为了展示给你,但我们不需要细节——除了为了与矢量版本进行比较,j μ表示电流密度或电磁场源:

With these definitions, the four beautiful Maxwell equations of chapter 8 become even more economical, even more elegantly simple. I’ll give them here just to show you, but we don’t need the details—except that for comparison with the vector version, jμ designates the current density or source of the electromagnetic field:

F μν = 4π j μ

F μν,λ + F νλ,μ + F λμ,ν = 0。

Fμν = 4πjμ

Fμν,λ + Fνλ,μ + Fλμ,ν = 0.

闵可夫斯基已经检查过F μν确实是我们现在所说的张量。18所做的非常出色:他将三维矢量EB(它们不是独立于框架的)转换为四维时空张量 F μν (例如,运动的电荷会产生磁场,因此B在电荷的静止框架中为零,但在相对运动的观察者框架中不为零它的分量在洛伦兹变换下会发生变化,但如果整个张量在一框架中为零(或非零),则它在所有框架中都为零(或非零)。

Minkowski had checked that Fμν is, indeed, what we now call a tensor.18 What he’d done was quite brilliant: he’d turned the 3-D vectors E and B, which are not frame independent—for instance, a moving electric charge creates a magnetic field, so B is zero in the charge’s rest frame but not in a relatively moving observer’s—into the 4-D space-time tensor Fμν, which is! Its components change under Lorentz transformations, but if the whole tensor is zero (or nonzero) in one frame, it is zero (or nonzero) in all of them.

但是,如果你浏览过第 11 章,你可能会想知道这对张量方程中指标的位置。但张量指标可以使用度量张量来提高或降低,因此在闵可夫斯基时空中,F μνF μν只是表达同一事物的不同方式。(不过,你需要在第一个方程中使用上层指标,以便对重复指标v求和。)

If you skimmed chapter 11, however, you might be wondering about the position of the indices in this pair of tensor equations. But tensor indices can be raised and lowered using the metric tensor, so in Minkowski space-time Fμν and Fμν are just different ways of saying the same thing. (You need an upstairs index in the first equation, though, in order to sum over the repeated index v.)

只是为了向你表明,这对美丽的方程式并不是张量优雅的定论,在后来的微分几何语言中,麦克斯韦方程组甚至更加简洁:

Just to show you that this beautiful pair of equations is not the last word on tensor elegance, in the later language of differential geometry, Maxwell’s equations are even more concise:

d * F =4π* J,dF =0。

d*F = 4π*J, dF = 0.

这里,符号从指标形式转回到海维赛德的粗体符号,表示整个向量,因此F是分量为F μν的张量。d是梯度算,类似于上面指标形式方程中偏导数的逗号。(星号表示张量的“对偶”。)

Here the notation has swung back from index form to Heaviside’s boldface notation for whole vectors, so that F is the tensor with components Fμν. The d is the gradient operator, analogous to the comma for partial derivatives in the index-form equations above. (The star denotes the “dual” of the tensor.)

但是,与向量一样,您仍然需要分量形式来进行计算!

As with vectors, though, you still need the component forms to do the calculations!

如今,电磁场张量F μν通常被称为麦克斯韦张量或法拉第张量。19爱因斯坦和格罗斯曼真正需要的是一种将产生引力场的物质和能量合并成一个张量的方法。但我们还没有完成电磁理论的研究。闵可夫斯基英年早逝后,他的朋友阿诺德·索末菲发展了他的思想,我们在第 9 章中看到。索末菲现在与列维-奇维塔和爱因斯坦都是朋友,而且他也对张量了解很多。因此,在 1910 年的论文“论相对论 I:四维矢量代数”中,他追随了闵可夫斯基的脚步,将与电磁场相关的基本物理量合并为一个张量T μν

Today, the electromagnetic field tensor Fμν is often called the Maxwell tensor or the Faraday tensor.19 What Einstein and Grossmann really needed, though, was a way to combine the matter and energy that produce a gravitational field into a single tensor, too. But we are not yet done with electromagnetic theory. After Minkowski’s premature death, his friend Arnold Sommerfeld developed his ideas as we saw in chapter 9. Sommerfeld was now friendly with both Levi-Civita and Einstein, and he also knew a lot about tensors. So, in his 1910 paper “On the Theory of Relativity I: Four-Dimensional Vector Algebra,” he had followed Minkowski’s lead and had combined the essential physical quantities associated with an electromagnetic field into a single tensor Tμν.

例如,已知平面电磁波在三维空间中携带的能量,矢量积E × B与单位面积能量流速成正比,即能量通量通量是流速,如图6.1所示)。由于E × B是另一个矢量,它有三个分量,索末菲将其标记为T 41T 42T 43。接下来,他将T 44等同于场的电磁能量密度(它的与携带能量的电磁波振幅的平方成正比)。他将其余成分与所谓的“麦克斯韦应力”联系起来。

For example, it was known that, for the energy carried through 3-D space by a plane electromagnetic wave, the vector product E × B is proportional to the rate of energy flow per unit area—that is, to the energy flux (flux is the rate of flow, as in fig. 6.1). Since E × B is another vector, it has three components, which Sommerfeld labeled T41, T42, T43. Next, he identified T44 with the electromagnetic energy density of the field (it’s proportional to the square of the amplitude of the electromagnetic wave carrying the energy). He identified the remaining components with what he called “Maxwell stresses.”

麦克斯韦已经将这些电动力和静电应力的作用等同于辐射压力——这是一个了不起的预测,在当今的许多应用中得到了证实,包括我在第 2 章中提到的光镊。由于压力需要力,而力是动量的变化,所以这些“应力”分量等同于动量通量。所以今天T μν通常被称为应力能量张量或能量动量张量。类似于图 9.4中的“麦克斯韦应力” ,这些指标告诉您哪个分量作用于小体积元素的哪个表面。例如,在笛卡尔坐标中,T 32将是作用于法线为y方向的表面的能量动量的z分量(反之亦然,因为事实证明T μν = T νμ 20)。

Maxwell had identified the effect of these electrokinetic and electrostatic stresses with radiation pressure—a remarkable prediction that is validated in many applications today, including the optical tweezers I mentioned in chapter 2. Since pressure requires a force and force is a change in momentum, these “stress” components are identified with momentum flux. So today Tμν is generally called the stress-energy tensor or the energymomentum tensor. Analogously with the “Maxwell stresses” in figure 9.4, the indices tell you which component is acting across which surface of a little volume element. For example, in Cartesian coordinates T32 would be the z-component of energy-momentum acting across the surface whose normal is in the y direction (and vice versa, for it turns out that Tμν = Tνμ20).

如此多的信息可以组合成一个张量,真是令人惊叹:EB组合成F μν,能量、物质和动量组合成T μν,当然还有引力场组合成g μν。这是张量在表示数据方面威力的另一个例子。但它也是爱因斯坦要求所有观察者都推导出相同物理现象的关键,因为里奇设计了张量来编码不变性。不幸的是,爱因斯坦和格罗斯曼在找出描述引力场的正确方程方面还有很长的路要走——即使坐标系发生变化,方程的形式也保持不变。

It’s quite amazing the way a lot of information can be combined into a single tensor like this: E and B into Fμν, energy, matter, and momentum into Tμν, and of course the gravitational field into gμν. It’s another example of the power of tensors in representing data. But it also holds the key to Einstein’s requirement that all observers deduce the same physics, because Ricci had designed tensors to encode invariance. Unfortunately, Einstein and Grossmann still had a way to go in figuring out the right equations to describe the gravitational field precisely—equations that kept their form even when the coordinate frame was changed.

ENTWURF:走向广义相对论之路

ENTWURF: ON THE ROAD TO GENERAL RELATIVITY

这两位朋友最初的想法是,他们需要一个张量方程,将相对论类似方程 ∇ 2 V = 4π G ρ 的左右两边联系起来——显然,索末菲电磁方程T μν的引力版本会出现在方程的右边。但爱因斯坦和格罗斯曼对使用里奇张量作为方程的左边犹豫不决。事实上,他们开始怀疑张量是否是合适的工具,因为无论他们如何努力,他们就是找不到合适的形式不变方程。

The two friends had started out with the idea that they needed a tensor equation, linking the left and right sides of the relativistic analogue of ∇2V = 4πGρ—and obviously a gravitational version of Sommerfeld’s electromagnetic Tμν would figure on the right-hand side. But Einstein and Grossmann were having second thoughts about using the Ricci tensor for the left-hand side. In fact, they’d begun to have doubts about whether tensors were the right tools at all, because no matter how hard they tried, they simply couldn’t find a suitably form-invariant equation.

这是因为他们同时处理许多不同的问题,而这些问题都必须融入最终的方程中。首先,这些方程必须简化为弱场(比如地球上的场)的牛顿极限,以及事件“局部”邻域中的狭义相对论,其中时空是“局部平坦的”。我们在图 10.1中看到了“局部平坦”的概念;正如你在下一个尾注中看到的那样,这个局部区域的大小随情况而变化。这些方程还必须产生已知物理定律(比如能量和动量守恒定律)的类似物,但这被证明特别困难——即使在今天,引力能量的概念仍然是有问题的。21这里的重点是,试图找到一个令人满意的相对论引力理论是一件多么复杂的事情!

This was because they were juggling many different balls that all had to meld into the final equations. For a start, these equations had to reduce to the Newtonian limit in weak fields such as that here on Earth—and to special relativity in the “local” neighborhood of an event, where the spacetime is “locally flat.” We saw the idea of “local flatness” in figure 10.1; how big this local region can be varies with the situation, as you can see in the next endnote. The equations also had to produce analogues of known physical laws such as conservation of energy and momentum, but this was proving particularly difficult—and even today the notion of gravitational energy is problematic.21 But the point here is what a complicated business it was, trying to find a satisfactory relativistic theory of gravity!

尽管如此,爱因斯坦和格罗斯曼还是在 1913 年发表了今天被称为Entwurf理论的论文——Entwurf在德语中是“大纲”的意思,论文标题的英文翻译是“广义相对论和引力理论大纲”。它包含了广义相对论的基本框架,这是一项了不起的成就,是数月辛勤工作的结晶。这些想法是爱因斯坦提出的,他撰写了论文的物理部分;格罗斯曼拥有数学专业知识,他撰写了数学部分。

Nonetheless, in 1913 Einstein and Grossmann published what is known today as the Entwurf theory—Entwurf is the German for “outline,” and the English translation of their paper’s title is “Outline of a Generalized Theory of Relativity and of a Theory of Gravitation.” It contained the essential framework of general relativity, and it was a remarkable achievement, the culmination of months of intense work. The ideas were Einstein’s, and he wrote the physics part of the paper; Grossmann had the mathematical expertise, and he wrote the mathematical section.

不幸的是,尽管Entwurf方程是张量的,但它们仅在某些受限的坐标变换下才是协变的(形式不变的)。然而,广义相对论的整个要点是应该允许任何参考系来表示各种相对运动的观察者。这就是格罗斯曼对张量的值产生怀疑的原因:广义协变在引力方面似乎是不可能的。爱因斯坦绞尽脑汁试图解释他的理论没有达到他最初的目标这一事实——而且“怀着沉重的心情”他也说服自己,更普遍的理论根本不可能存在。22

Unfortunately, although the Entwurf equations were tensorial, they were only covariant—form invariant—under certain restricted coordinate transformations. Yet the whole point of general relativity was that any reference frame should be allowed, to represent all sorts of relatively moving observers. That’s why Grossmann had doubts about the value of tensors: general covariance didn’t seem possible when it came to gravity. Einstein tied himself up in knots trying to explain away the fact that his theory didn’t live up to his initial goal—and “with a heavy heart” he, too, convinced himself that a more general theory simply wasn’t possible.22

1914 年,爱因斯坦和格罗斯曼发表了第二篇联合论文,以更优雅的方式推导出了Entwurf引力方程,但它们仍然不是完全协变的。爱因斯坦仍在继续尝试。自 1907 年以来,他发表了近十几篇关于他探索相对论引力理论的论文,而且还有更多论文即将面世。当一切结束后,他会开玩笑地对一个朋友说:“爱因斯坦过得很轻松,每年他都会撤回前一年写的东西。”最后,他每周都会撤回或更新文章。然而,在 1914 年,他还没有笑出来,因为他的同事对Entwurf理论的反应并不太热烈。他向老朋友米歇尔·贝索抱怨道:“物理学家们表现得相当被动……亚伯拉罕似乎对它有最深刻的理解。可以肯定的是,他猛烈抨击所有相对论……但他是在理解的基础上这样做的。”爱因斯坦接着说,量子理论的共同创始人、爱因斯坦和狭义相对论的早期拥护者马克斯·普朗克是那些“不接受”他的新理论的人之一,但他希望索末菲和洛伦兹能接受。23

Einstein and Grossmann published a second joint paper in 1914, deriving the Entwurf gravitational equations in a more elegant way, but still they weren’t fully covariant. And still Einstein kept trying. Since 1907 he’d published almost a dozen papers on his journey to a relativistic theory of gravity, and there were more to come. When it was all over, he would joke to a friend, “Einstein has it easy, every year he retracts what he wrote the year before.” By the end he’d be retracting or updating every week. In 1914, though, he wasn’t yet laughing, for his colleagues reacted none too warmly to the Entwurf theory. He complained to his old friend Michele Besso, “The fraternity of physicists behaves rather passively.... Abraham seems to have the greatest understanding for it. To be sure, he fulminates against all relativity ... but he does it with understanding.” Einstein went on to say that Max Planck—cofounder of quantum theory, and an early champion of Einstein and special relativity—was one of those who were “not open” to his new theory, but he hoped Sommerfeld and Lorentz might be.23

洛伦兹一直很感兴趣并支持他,但他从未真正放弃以太的想法。索末菲也是一位值得尊敬的同事——他谨慎地接受了设计方法,后来他甚至建议爱因斯坦确保自己是新理论的主要作者。但爱因斯坦对他的老朋友很有信心,他回答说:“格罗斯曼永远不会声称自己是共同发现者”;爱因斯坦说,格罗斯曼的角色是“引导”他学习数学。此后,爱因斯坦确实证明了自己能够创新地应用张量数学。24

Lorentz was always interested and supportive, but he never really gave up the idea of the ether. Sommerfeld, too, was a valued colleague—he cautiously accepted the Entwurf approach, and later he even advised Einstein to make sure he was known as the main author of the new theory. But Einstein had confidence in his longtime friend, replying, “Grossmann will never claim to be co-discoverer”; Grossmann’s role, Einstein said, had been “to orient” him mathematically. After that, Einstein did indeed prove more than capable of innovatively applying tensor maths.24

• • •

• • •

尽管爱因斯坦正在努力研究他的新引力理论,但他总是能抽出时间与他的新博士后助理奥托·斯特恩交流。我们在发现量子自旋时认识了斯特恩。当他第一次来和爱因斯坦一起工作时,斯特恩惊讶地发现“一个不打领带的人坐在办公桌后面”,看起来更像一个“修路工”,而不是他所期待的名人。但他说,爱因斯坦“非常好” 。25

Although Einstein was working hard on his new theory of gravity, he always found time for his new postdoctoral assistant Otto Stern, whom we met in connection with the discovery of quantum spin. When he’d first turned up to work with Einstein, Stern had been surprised to find “a guy without a tie sitting behind a desk” and looking more like a “road-mender” than the famous man he’d expected. But, he said, Einstein “was terribly nice.”25

爱因斯坦仍然抽出时间与好朋友相处,比如贝索,他于 1913 年 6 月访问了苏黎世。这也很好,因为这次访问很有成果。爱因斯坦已经想出了一种测试Entwurf理论的方法,看看它是否能解释水星近日点进动中缺失的 43 角秒——贝索和他一起计算。不幸的是,他们只能找到 18 秒,而不是 43 秒。不过,正如我们将看到的,爱因斯坦后来成功地使用他和贝索设计的方法——这也许就是为什么 2022 年,他们最初的计算,即所谓的爱因斯坦-贝索手稿,在拍卖会上以超过 1500 万美元的价格售出。

Einstein still found time for good friends, too, such as Besso, who visited Zurich in June 1913. Which was just as well because it turned out to be a fruitful visit. Einstein had already figured out a way to test the Entwurf theory, by seeing if it could account for the missing 43 angular seconds in the precession of the perihelion of Mercury—and Besso joined him in working out the calculations. Unfortunately, they could only manage to find 18 seconds, not 43. As we’ll see, though, Einstein would later successfully use the method he and Besso devised—which may be why, in 2022, their original calculations, in the so-called Einstein-Besso Manuscript, sold at auction for more than $15 million.

贝索比爱因斯坦更早地发现了“设计论”的一些问题,而不仅仅是它没有正确预测水星的位置。但爱因斯坦还有其他问题需要解决:1914 年,他接受了马克斯·普朗克提供的柏林教授职位——鉴于当时远距离通信的困难,这一举动结束了他与格罗斯曼的密切合作——他与米列娃·马里奇的婚姻最终破裂。部分原因是爱因斯坦的工作。“你看,他这么有名,留给妻子的时间不多了,”几年前马里奇在写给一位朋友的信中写道。她接着说,她意识到自己“摆出一副傲慢自大的架势”,但她不知道那是因为害羞还是骄傲。她所知道的是,“我非常渴望爱情。”爱因斯坦没有像在幸福的年代那样试图打破她勇敢沉默的外表。相反,他与住在柏林的表妹艾尔莎变得亲近;他们于 1919 年结婚。26

Besso could see some of the problems with the Entwurf theory before Einstein did—and not just the fact that it didn’t give the right prediction for Mercury. But Einstein had other problems to contend with: in 1914, he took up Max Planck’s offer of a professorship in Berlin—a move that ended his close collaboration with Grossmann, given the difficulties of long-distance communication in those days—and his marriage to Mileva Marić finally fell acrimoniously apart. Part of the reason was Einstein’s work. “You see, with such fame, not much time remains for his wife,” Marić had written to a friend a couple of years earlier. She went on to say that she realised she “puts on a haughty and superior air,” but she didn’t know if that was because of shyness or pride. What she did know was that “I am very starved for love.” Einstein had not tried to penetrate her brave, taciturn front, as he might have done in happier times. Instead, he had become close to his cousin Elsa, who conveniently lived in Berlin; they would marry in 1919.26

当然,还有第一次世界大战,从 1914 年持续到 1918 年。作为一名和平主义者,爱因斯坦避开了这场战争,尽管他现在在柏林,但他拒绝签署德皇的“致文化世界宣言”,该宣言否认了德国对战争的责任。马克斯·普朗克和费利克斯·克莱因是众多在宣言上签名的受人尊敬的德国科学家之一,但大卫·希尔伯特也拒绝签名。他是闵可夫斯基的老朋友,也是克莱因的学生,现在他是哥廷根大学的教授,他帮助哥廷根成为一个著名的充满活力的智力中心。希尔伯特是一位热爱跳舞的数学天才,但他也是一个离谱的调情者——很像爱因斯坦。但他们都坚定地坚持反战原则,而克莱因——他是如此爱国,却没有阅读文件就签署了文件——后来后悔如此草率行事。27

Then, of course, there was the First World War, from 1914 into 1918. As a pacifist, Einstein steered clear of it, and although he was now in Berlin, he refused to sign the kaiser’s “Declaration to the Cultural World,” which disclaimed Germany’s responsibility for the war. Max Planck and Felix Klein were among the many respected German scientists who added their signatures to the declaration, but David Hilbert also refused to sign. He’d been Minkowski’s longtime friend, and a student of Klein, and he was now a professor at Göttingen, which he’d helped become a famously vibrant intellectual centre. Hilbert was a mathematical genius who loved dancing, but who was also an outrageous flirt—rather like Einstein. But they both held firm to their antiwar principles, and Klein—who’d signed the document without reading it, so patriotic was he—came to regret acting so hastily.27

战争还打乱了爱因斯坦测试其理论的另一项计划。1911 年,他预测了光线经过太阳时会发生弯曲的角度,并联系了天文学家,看看这种弯曲是否能在日食期间无法探测到。从地球上看时靠近太阳的遥远恒星在日全食期间可见,它们的位置可以被拍照并与几个月后相同恒星的位置进行比较,那时从地球到这些特定恒星的视线不会经过太阳附近。(恒星本身距离地球太远,以至于在那几个月里,它们相对于地球的实际位置实际上保持不变。)几次计划中的日食探险被放弃或因战争而严重中断。具有讽刺意味的是,这对爱因斯坦来说是幸运的:就像他第一次预测水星一样,他在 1911 年对光线的预测被证明太小了。

The war also disrupted another of Einstein’s plans to test his theory. In 1911, he’d predicted the angle at which light would be bent as it passed by the Sun, and he’d approached astronomers to see if this bending could be detected during a solar eclipse. Distant stars that appeared close to the Sun when viewed from Earth would become visible during a total eclipse, and their positions could be photographed and compared with the same stars’ positions months later, when the line of sight from Earth to those particular stars did not pass near the Sun. (The stars themselves are so far away that during those few months, their actual positions relative to Earth remain effectively unchanged.) Several planned eclipse expeditions were abandoned or fatally disrupted by the war. Ironically, this was lucky for Einstein: like his first Mercury prediction, his 1911 light prediction would prove too small.

图里奥·莱维-西维塔和大卫·希尔伯特上

ENTER TULLIO LEVI-CIVITA AND DAVID HILBERT

1914 年 11 月,爱因斯坦发表了一篇论文,进一步发展和完善了他与格罗斯曼合作的工作——他立即将副本寄给了他的劲敌马克斯·亚伯拉罕。亚伯拉罕则立即写信给他的朋友列维-奇维塔,告诉他他不理解爱因斯坦的推理,并建议他和列维-奇维塔一起讨论。两人在帕多瓦见面,列维-奇维塔对他们的讨论感到非常兴奋,以至于在 1915 年初,他第一次联系了爱因斯坦。他想澄清爱因斯坦使用张量的一些错误,爱因斯坦感激地回复道:“你如此仔细地检查我的论文,真是帮了我大忙。你可以想象,很少有人能独立、批判地深入研究这个主题。” 28

In November 1914, Einstein published a paper further developing and polishing the work he’d done with Grossmann—and he immediately sent a copy to his worthy opponent Max Abraham. Abraham, in turn, promptly wrote to his friend Levi-Civita, telling him he didn’t understand Einstein’s reasoning, and suggesting that he and Levi-Civita get together to discuss it. The two men met in Padua, and Levi-Civita was so excited by their discussions that in early 1915, he reached out to Einstein for the first time. He wanted to clarify some errors in Einstein’s use of tensors, and Einstein replied gratefully: “By examining my paper so carefully, you are doing me a great favour. You can imagine how rarely someone delves independently and critically into this subject.”28

在接下来的两个月里,他们每人交换了十几封信,爱因斯坦一直在努力寻找正确的引力场张量。当意大利与德国开战时,帕多瓦和柏林之间的交流变得困难,通信也断断续续。29尽管如此,1915 年夏天,当爱因斯坦应希尔伯特的邀请前往哥廷根发表六场关于引力的系列讲座时,他还是感到很有信心。到这个阶段,他对张量的掌握已经比大多数数学家都要好,因为里奇的工作仍然默默无闻。

Over the next two months they exchanged some dozen letters each as Einstein struggled to find the right gravitational field tensor. When Italy entered the war, against Germany, communication between Padua and Berlin became difficult, and the correspondence faltered.29 Still, in the summer of 1915 Einstein was feeling confident when he went to Göttingen, at Hilbert’s invitation, to deliver a series of six lectures on gravity. By this stage, he had mastered tensors better than most mathematicians, for Ricci’s work was still languishing in the shadows.

爱因斯坦的自信并非毫无根据。36 岁时,他很高兴自己的思想得到了数学界的元老的认可政治家克莱因和希尔伯特——克莱因现在快 70 岁了,希尔伯特 53 岁。他们都是不变理论的专家,希尔伯特还开创了几何学的公理化方法(就像朱塞佩·皮亚诺对代数所做的那样)。希尔伯特似乎对爱因斯坦的几何(度量张量)引力方法印象特别深刻,鼓励他不要放弃对协变引力场方程的最初探索。

Einstein’s confidence was not misplaced. At thirty-six he was delighted with the reception his ideas received from the mathematical elder statesmen Klein and Hilbert—Klein was almost seventy now, and Hilbert was fifty-three. They both were experts in invariant theory, and Hilbert had also initiated an axiomatic approach to geometry (the way Giuseppe Peano had done for algebra). It seems that Hilbert, who was especially impressed with Einstein’s geometrical (metric tensor) approach to gravity, encouraged him not to give up his original search for covariant gravitational field equations.

爱因斯坦放弃广义协变性思想的原因之一是,它意味着度量的形式在任何坐标变换下都应该是不变的;但这种普遍性将包括与相对运动无关的变换——例如从笛卡尔坐标系到极坐标系的变换,其中度量的形式不是不变的。这类似于分别在笛卡尔坐标系和极坐标系中表达圆的方程:x 2 + y 2 = a 2r = a。这是同一个圆,但从表面上看,这两个方程看起来完全不一样。爱因斯坦最终意识到,即使有这些类型的变换,也可以找到广义协变方程——假设变换是齐次的。(这意味着在一个框架中为零的张量在所有框架中都是零,爱因斯坦意识到这提供了一种编写不变张量方程的方法。)30

One of the reasons that Einstein had abandoned the idea of general covariance was that it meant the form of the metric should be invariant under any coordinate transformation; but this kind of generality would include transformations that had nothing to do with relative motion—such as transforming from Cartesian to polar coordinates, where the form of the metric isn’t invariant. It’s analogous to expressing the equation of a circle in Cartesian and polar coordinates, respectively: x2 + y2 = a2 and r = a. It’s the same circle, but on the face of it, the equations look nothing alike. Einstein would eventually realise that even with these kinds of transformations, generally covariant equations could be found—assuming the transformations are homogeneous. (This means that tensors that are zero in one frame are zero in all frames, and Einstein realised that this gives a way to write invariant tensor equations.)30

回到柏林后,爱因斯坦继续坚持不懈地研究他的方程。正如我之前提到的,他试图将能量动量张量T μν与由g μν及其二阶导数构成的合适张量表达式联系起来——类似于牛顿方程 ∇ 2 V = 4π G ρ。经过数月的绞尽脑汁,试图找到正确的g μν表达式后,爱因斯坦突然意识到Entwurf理论走错了路。当他抛弃黎曼和里奇张量时,他误解了该理论的牛顿极限,但现在他再次审视了他和格罗斯曼三年前放弃的工作。因为黎曼和里奇张量是由他需要的g μν的导数构成的。但当他研究新方法时,他收到了索末菲的坏消息:希尔伯特正在偷猎他的保护区。至少爱因斯坦是这么认为的。

Back in Berlin, Einstein kept working doggedly on his equations. As I mentioned earlier, he was trying to link the energy-momentum tensor Tμν with a suitable tensor expression made from the gμν and its second derivatives—by analogy with the Newtonian equation ∇2V = 4πGρ. After wracking his brains for months, trying to find the right gμν expression, Einstein suddenly realised the Entwurf theory was on the wrong track. He had misunderstood the Newtonian limit of the theory when he threw aside the Riemann and Ricci tensors, but now he looked again at the work he and Grossmann had discarded three years earlier. For the Riemann and Ricci tensors are made from the very derivatives of gμν that he needed. But as he worked on his new approach, he received unwelcome news from Sommerfeld: Hilbert was poaching on his preserves. At least, that’s how Einstein saw it.

在爱因斯坦访问哥廷根之前,希尔伯特对古斯塔夫·米 1912 年提出的物质电磁理论很感兴趣。米是德国东北部格赖夫斯瓦尔德大学的物理学教授,他 1908 年提出的球形粒子电磁辐射散射理论至今仍被广泛引用。后来,他试图寻找物质和电磁辐射的统一理论,但希尔伯特当时对此很感兴趣。爱因斯坦对米理论中的引力分析并不感兴趣,因此希尔伯特在听了爱因斯坦在哥廷根的讲座后,决定尝试将米的电磁学方法与爱因斯坦的引力几何方法结合起来。他追随米的脚步,认为物质的起源是电磁的,由于物质产生引力,希尔伯特认为他可以改进米统一电磁学和引力的尝试。

Before Einstein’s Göttingen visit Hilbert had been interested in Gustav Mie’s 1912 electromagnetic theory of matter. Mie was a physics professor at the University of Greifswald in northeastern Germany, and his 1908 theory of the scattering of electromagnetic radiation by a spherical particle is still much cited. Not so his later attempt to find a unified theory of matter and electromagnetic radiation, but at the time, Hilbert was much taken with it. Einstein had been unimpressed with the gravitational analysis in Mie’s theory, and so Hilbert, after hearing Einstein’s lectures at Göttingen, decided to try to mesh Mie’s approach to electromagnetism with Einstein’s geometric approach to gravity. Following Mie, he believed that matter was electromagnetic in origin, and since matter produced gravity, Hilbert thought he could improve Mie’s attempt to unify electromagnetism and gravity.

爱因斯坦在相对论上奋斗了十年,已经筋疲力尽,但也许想到希尔伯特将要继续他的引力理论,他又继续前行。如果在过去十年中奠定了所有基础的希尔伯特比他先冲过终点线,那将令人心碎。无论如何,他在 1915 年 11 月全程努力工作,一路与希尔伯特交流进展——可能是因为他想表明自己的优先权,也是为了分享想法并传达令人兴奋的进展。(似乎这还不够累,与此同时,他还在给前妻和 11 岁的儿子汉斯·阿尔伯特写信——他的小儿子爱德华只有五岁,还不够写信,但爱因斯坦希望与他们建立更好的关系。)31

Einstein was exhausted after a decade of hard work on relativity, but perhaps the thought that Hilbert was taking up his gravitational ideas spurred him on. It would be heartbreaking if, having laid all the groundwork over the past decade, Hilbert rode across the finishing line just ahead of him. At any rate, he worked ferociously hard throughout November 1915, exchanging updates with Hilbert along the way—possibly because he wanted to stake his priority, but also to share ideas and to convey exciting progress. (As if this weren’t taxing enough, at the same time he was also writing to his ex-wife and eleven-year-old son, Hans Albert—his younger son, Eduard, was only five, not yet old enough for letters, but Einstein hoped to build better relationships with them all.)31

具体来说,他告诉希尔伯特他将于 11 月 18 日向普鲁士科学院提交一篇论文,在论文中他将他最新的方程应用于水星的近日点。这一次,他找到了所有 43 个缺失的角秒——这一结果让他“欣喜若狂了好几天”,他后来告诉一位朋友。当然,他立即把这个消息发给了贝索,贝索帮助他设计了找到这一宏伟结果的方法:“定量解释近日点的运动”,他得意地写道。32事实上,整个 11 月,爱因斯坦在四周内写了四篇论文,每一篇论文都离最终的、完整的论文更近了一步11 月 25 日,他怀着无比激动和欣慰的心情提交了协变的广义相对论。

In particular, he told Hilbert about a paper he was presenting to the Prussian Academy of Sciences on November 18, in which he applied his latest equations to Mercury’s perihelion. This time he’d found all 43 of those missing angular seconds—a result that made him “beside [him]self with ecstasy for days,” as he later told a friend. He’d immediately sent the news to Besso, of course, who’d helped him devise the method he used to find this magnificent result: “Motions of the perihelion quantitatively explained,” he wrote triumphantly.32 Indeed, throughout November Einstein wrote four papers in four weeks, each paper getting closer to the final, fully covariant, general theory of relativity that he submitted, with enormous excitement and relief, on November 25.

您可能已经听说了接下来发生的事情,所以在我们看到爱因斯坦著名方程之前,让我们先把优先权问题解决掉。在爱因斯坦 11 月最后一篇论文的五天前,希尔伯特向哥廷根科学院提交了他自己的论文“物理学基础(第一篇笔记)”——这篇论文显然也包含了正确的引力场方程。自 20 世纪 20 年代以来,有人一直混淆爱因斯坦方程应该称为希尔伯特方程还是爱因斯坦-希尔伯特方程。这种混淆在 20 世纪 90 年代末引发了争议,当时特拉维夫大学的科学史学家 Leo Corry 发现希尔伯特 11 月 20 日论文的原始页面校样上印有 12 月 6 日的印刷日期戳——晚于爱因斯坦 11 月 25 日论文发表的 12 月 2 日。更重要的是,这些证明似乎根本不包含爱因斯坦方程——尽管在米氏电磁学的受限情况下,它们确实包含一个隐含等效的公式。然而,与爱因斯坦不同,希尔伯特证明中的基础理论不是完全协变的。这一点,加上显式场方程确实出现在希尔伯特 11 月 20 日论文的后续出版版本中,并附有对爱因斯坦 11 月四篇论文的致谢,导致几位专家得出结论,希尔伯特只是从爱因斯坦那里学到了方程的显式形式。33

You may have heard what happened next, so before we see Einstein’s famous equations, let’s get the issue of priority out of the way. Five days before Einstein’s final November paper, Hilbert submitted his own paper, “Foundations of Physics (first note),” to the Göttingen Academy of Sciences—a paper that apparently also contained the right gravitational field equations. And ever since the 1920s, there has been confusion in some quarters over whether the Einstein equations should, in fact, be called the Hilbert equations, or the Einstein-Hilbert equations. The confusion became controversy in the late 1990s, when science historian Leo Corry, of Tel Aviv University, discovered that the original page proofs of Hilbert’s November 20 paper bore the printer’s date stamp of December 6—after December 2 when Einstein’s November 25 paper was published. What’s more, these proofs did not appear to contain the Einstein equations at all—although they do contain a formulation that is implicitly equivalent, in the restricted case of Mie’s electromagnetism. Unlike Einstein’s, though, the underlying theory in Hilbert’s proofs was not fully covariant. This, together with the fact that the explicit field equations do appear in the later published version of Hilbert’s November 20 paper, with acknowledgments to Einstein’s four November papers, has led several specialists to conclude that Hilbert only learned the explicit form of the equations from Einstein.33

另一方面,证明中有一页缺失了,而这一页可能包含具体的方程式。这为阴谋论提供了完美的素材——谁撕毁了这一页?——双方的一些人都用激烈的言辞进行争论。不过,据我所知,大多数学者认为缺失部分的上下文表明希尔伯特几乎肯定没有在“爱因斯坦方程式”上击败爱因斯坦。此外,每个人都同意广义相对论是爱因斯坦的,如果没有爱因斯坦坚实的基础,就不会有“希尔伯特方程式” 。34

On the other hand, there is part of a page missing from the proofs, which may have contained the specific equations. Perfect fodder for conspiracy theories—who tore the page?—and intemperate language has been traded by some on both sides of the debate. As far as I can tell, though, most of these scholars believe the context of the missing section indicates that Hilbert almost certainly didn’t beat Einstein to the “Einstein equations.” Besides, everyone agrees that general relativity is Einstein’s, and that there would be no “Hilbert equations” without Einstein’s solid foundations to build on.34

我在尾注中引用了许多这样的学术论文,35但无论如何,似乎玩家之间并没有真正争论谁找到了广义相对论的最终方程——而是其他人,包括克莱因,几年后也注意到他们发现的时间相似。36当然,希尔伯特从未声称自己是广义相对论的共同作者,尽管如果爱因斯坦没有先于他发表论文,他可能是。37事实上,当他的论文最终发表时,他不仅引用了爱因斯坦 11 月的论文,还经常称赞爱因斯坦。“哥廷根街头的每个男孩都比爱因斯坦更了解四维几何,”他曾宣称;“然而,尽管如此,这项工作是爱因斯坦而不是数学家们完成的。”在最初感到希尔伯特窃取了他的想法之后,到 1915 年底,爱因斯坦又与他建立了友谊。38

I’ve referenced many of these scholarly papers in the endnote,35 but either way, it seems there was no real dispute between the players themselves about who found the final equations of general relativity—it was others, including Klein, who, several years later, drew attention to the similar timing of their discoveries.36 Certainly Hilbert never claimed to be the coauthor of general relativity, although possibly he might have, had not Einstein published ahead of him.37 As it is, he not only cited Einstein’s November papers when his own paper was eventually published, he often praised Einstein, too. “Every boy in the streets of Göttingen understands more about four-dimensional geometry than Einstein,” he once proclaimed; “yet, in spite of that, Einstein did the work and not the mathematicians.” And after that initial bitter feeling that Hilbert had stolen his ideas, by the end of 1915 Einstein had reached out to him in friendship again.38

这个故事真正表明的是,即使是最伟大的天才也需要彼此的启发——爱因斯坦是当代最伟大的物理学家,他无疑受益于希尔伯特的数学严谨性,而希尔伯特是当代最优秀的数学家,他无疑受益于爱因斯坦的物理洞察力。但正是爱因斯坦一人建立了广义相对论的基础,并在格罗斯曼的最初帮助下,使张量分析名声大噪。那么让我向你展示他最终是如何做到的。

What this story really shows is that even the greatest geniuses need inspiration from each other—Einstein the greatest living physicist, who no doubt benefited from Hilbert’s mathematical rigour, and Hilbert the best living mathematician, who certainly benefited from Einstein’s physical insight. But it is Einstein alone who built the foundations of general relativity, and with Grossmann’s initial help, put tensor analysis on the map. So let me show you how he finally did it.

爱因斯坦终于提出了广义相对论

EINSTEIN’S GENERAL THEORY OF RELATIVITY, AT LAST

我们之前看到,索末菲效仿闵可夫斯基,创建了一个张量T μν,它包含了电磁场携带的能量和动量的所有基本信息。事实证明,T μν的散度具有非常特殊的性质。时空中的“散度”的定义类似于我们在图 7.1中看到的普通矢量微积分运算,其中矢量的散度=++++。它是一个标量,所以它是不变的——对任何坐标都是正确的。所以,如果我们使用里奇符号xx 1XV 1等等——以及偏导数的逗号符号、汉密尔顿的 nabla 和爱因斯坦求和约定(其中重复的上下指标表示和)——普通的向量微积分散度可以表示如下:

We saw a little earlier that, following Minkowski, Sommerfeld had created a single tensor, Tμν, which contained all the essential information about the energy and momentum carried by an electromagnetic field. It turns out that the divergence of Tμν has a very special property. “Divergence” in space-time is defined by analogy with the ordinary vector calculus operation that we saw in figure 7.1, where the divergence of a vector V=Xi+Yj+Zk is Xx+Yy+Zz. It’s a scalar, so it is invariant—true for any coordinates. So, if we use Ricci’s notation xx1, XV1, and so on—as well as the comma notation for partial derivatives, Hamilton’s nabla, and the Einstein summation convention (where repeated up and down indices indicate a sum)—the ordinary vector calculus divergence can be represented like this:

  =  11 +22 + 33   ν ν

 V = V 1,1 +V2,2 +V 3,3  V ν ,ν.

将其扩展到闵可夫斯基时空,我们可以使用相同的表示形式V ν,其中指标现在范围从 1 到 4(如果时间坐标用 0 表示,则为 0 到 3)。39是闵可夫斯基时空中散度的定义,类比于欧几里得向量定义——现在将类比扩展到这个平坦的闵可夫斯基时空中的张量,并将能量动量张量的散度定义为T μ ν。(正如我所提到的,使用度量可以提高或降低张量的指标,因此T μνT μν是书写相同信息的不同方式。)但是,无论你想如何称呼这个散度表达式,事实证明,对于电磁学以及许多其他物理情况,当你进行计算时,你总是会得到这个方程:

Extending this to Minkowski space-time, we can use the same representation, V ν, where the indices now range from 1 to 4 (or 0 to 3 if the time coordinate is denoted with a 0).39 This is a definition of divergence in Minkowski space-time, by analogy with the Euclidean vector definition—so now extend the analogy to tensors in this flat, Minkowskian space-time, and define the divergence of the energy-momentum tensor to be Tμν. (As I’ve mentioned, with a metric you can raise or lower the indices of a tensor, so Tμν and Tμν are different ways of writing the same information.) Regardless of what you want to call this divergence expression, though, it turns out that for electromagnetism, and for many other physical situations, when you do the calculations, you always get this equation:

Tμν , ν =0。

Tμν = 0.

事实证明,当你考虑这四个方程的含义时(因为 μ 在时空中取四个值,所以是四个),你会得到当 μ = 4(或 0,无论你喜欢哪个时间分量的指标)时的能量守恒定律,加上构成通常的动量守恒定律的三个空间分量方程。换句话说,能量和动量守恒定律通过能量动量张量T μν被整合成一个看起来非常简单的方程。这是信息以张量形式表示的又一个非常经济的例子。

It also turns out that when you consider the meaning of each of these four equations—four because μ takes four values in space-time—you get the law of conservation of energy when μ = 4 (or 0, whichever index you prefer for the time component), plus the three spatial component equations making up the usual law of conservation of momentum. In other words, the laws of conservation of energy and momentum are rolled into one beautifully simple-looking equation via the energy-momentum tensor Tμν. It’s another example of the brilliantly economical way information is represented in tensors.

自 1907 年以来,爱因斯坦一直在努力解决引力能量守恒问题。问题并不在于将什么放入T μν中,因为这将根据所考虑的物质和能量分布的特征来定义——就像索末菲将电磁场的已知特征放入他的T μν中一样。问题在于守恒定律。现在是时候回答爱因斯坦在与格罗斯曼合作之初提出的问题了:如何将物理定律从狭义理论转移到广义理论。

Einstein had been grappling with the conservation of gravitational energy since 1907. It wasn’t so much about what to put into Tμν, for that would be defined from the features of the matter and energy distribution under consideration—just as Sommerfeld put the known features of an electromagnetic field into his Tμν. The problem was with the conservation law. So now it’s time to answer the question Einstein had asked at the beginning of his collaboration with Grossmann: how to transfer the laws of physics from the special theory to the general one.

张量守恒定律T μν =0 是狭义相对论中物理定律的一个例子,这意味着它对所有以恒定相对运动移动的观察者都保持相同的形式。换句话说,它在洛伦兹变换下是形式不变的(协变)。事实证明,要将洛伦兹协变张量方程改为广义协变方程,只需将偏导数替换为协变导数(我在第11 章末尾附近定义了它)。这个定义是一个体现符号力量的绝妙例子,它意味着你只需将上述方程中的所有逗号改为分号(类似地,我们之前看到的麦克斯韦方程中的逗号,以张量F μν的形式表示)。40所以,在广义相对论中,能量动量方程的(局部)守恒定律为T μν = 0。

The tensorial conservation equation Tμν =0 is an example of a physical law that holds in special relativity, which means it keeps the same form for all observers moving with constant relative motion. In other words, it is form-invariant (covariant) under Lorentz transformations. It turns out that all you have to do to change a Lorentz-covariant tensor equation to a generally covariant one is to replace partial derivatives by covariant derivatives (which I defined near the end of chap. 11). In a marvelous example of the power of symbolism, this definition means you simply have to change all the commas in the above equations to semicolons (and similarly for the commas in Maxwell’s equations that we saw earlier, written in terms of the tensor Fμν).40 So, in general relativity the (local) law of conservation of energy-momentum equation is Tμν = 0.

实际上,爱因斯坦不知道这条规则,也不使用这种分号符号——张量分析的许多细节都是后来才出现的;但撇开符号不谈,他还是得到了结果T μν =0。(你可以在方框中看到他写这个方程的方式。)

Actually, Einstein didn’t know this rule, or use this semicolon notation—many of the niceties of tensor analysis came later; but notation aside, he obtained the result Tμν =0 anyway. (You can see the way he wrote this equation in the box.)

将爱因斯坦守恒定律符号转化为现代形式

TRANSLATING EINSTEIN’S NOTATION FOR THE CONSERVATION EQUATION INTO THE MODERN FORM

爱因斯坦在 1916 年的综述《广义相对论基础》中,在他的方程 (57a) 中,以等效形式写出了能量动量守恒定律方程:

In his 1916 overview, The Foundation of the Theory of General Relativity, in his equation (57a), Einstein wrote the conservation of energy-momentum equation in the equivalent form:

δ电视σαδ一个=Γασβ电视βα

δTσαδxa=ΓασβTβα.

他用 α 代替我的 ν 来表示和,但这是一个任意的选择,就像他选择 σ 和我选择 μ 来标记T一样:正如我在第 11 章中提到的那样,任意向量和张量分量的标签本身也是任意的,就像我们在代数中用x表示未知数一样。不过,正如我们将在下面看到的,爱因斯坦在他的最终方程中使用了T μν,这就是我选择 μ 和 ν 作为标签的原因。

He’s used α instead of my ν to indicate the sums, but that’s an arbitrary choice, as is his choice of σ and my choice of μ to label T: as I mentioned in chapter 11, the labels on arbitrary vector and tensor components are themselves arbitrary, just like our time-honoured use of x for the unknown in algebra. As we’ll see below, though, Einstein used Tμν in his final equations, which is why I’ve chosen μ and ν for my labels.

这个等式右边的项是里奇 (Ricci) 加到普通偏导数上的曲率部分,用来定义协变导数。(Γ 符号是克里斯托费尔 (Christoffel) 符号。)因此,在两边都加上这个项,并使用分号符号来定义协变导数,爱因斯坦方程就变成了:

The term on the right-hand side in this equation is the curvature part Ricci added to the ordinary partial derivative to define a covariant derivative. (The Γ symbol is the Christoffel symbol.) So, adding this term to both sides, and using semicolon notation to define the covariant derivative, Einstein’s equation here becomes:

T α σ;α = 0。

Tασ;α = 0.

根据第 11 章的最后一课,你可以使用度量张量g σγ T α σ;α = T αγ ; α来提高和降低这里的指标,并且由于这个张量的指标是对称的,所以爱因斯坦写的完全等同于现代表示:

With one final lesson from chapter 11, you can raise and lower indices here with the metric tensor—g σγTασ;α = Tαγ —and since this tensor is symmetric in its indices, then what Einstein wrote is entirely equivalent to the modern representation:

电视αγα=电视γαα=0电视μ=0

Tαγ;α=Tγα;α=0Tμv;v=0,

其中等价性 ≡ 通过重新标记指标而来。

where the equivalence ≡ comes from relabeling the indices.

然而,由于各种复杂的原因,这条守恒定律似乎与他的场方程不相符,正如我们之前通过类比牛顿引力定律所看到的,他的场方程将表达式左侧使用g μν的二阶导数与右侧使用T μν联系起来。1915 年 11 月,当爱因斯坦最终将左侧的 R μν 换算为里奇张量时,他发现仅仅写成R μν = kT μν(其中k是比例常数)是不够的。这是因为当他取两边的散度时,他最终得到了一个对引力场施加了物理上不切实际的限制的方程。我们将在下一章中对此进行更多介绍。

Yet for various complicated reasons, this conservation law didn’t seem to gel with his field equations, which—as we saw earlier by analogy with the Newtonian law of gravity—related an expression using second derivatives of gμν on the left-hand side to Tμν on the right-hand side. In November 1915, when Einstein finally returned to the Ricci tensor for the left-hand side, he found it wasn’t enough simply to write Rμν = kTμν (where k is the proportionality constant). That’s because when he took the divergence of both sides, he ended up with an equation that placed physically unrealistic restrictions on the gravitational field. We’ll see more on this in the next chapter.

最终,到 1915 年 11 月 25 日,爱因斯坦发现了正确的引力场方程,他将其写成如下形式

Finally, by November 25, 1915, Einstein had discovered the correct gravitational field equations, which he wrote in the form

Rμ=电视μ12μ电视

Rμv=kTμv12gμvT,

在哪里电视=电视μμ,它是一个标量——因为上、下索引相同,表示分量相加,得到一个标量,正如我们在第 11 章中看到的那样——单位通常选择为k = 8π。41今天,爱因斯坦的方程通常写成等效42形式

where T=Tμμ, which is a scalar—because the up and down indices are the same, indicating the components are summed, giving a scalar as we saw in chapter 11—and units are usually chosen so that k = 8π.41 Today Einstein’s equations are often written in the equivalent42 form,

Rμ12μR=电视μ

Rμv12gμvR=kTμv.

这是希尔伯特在 11 月 20 日发表的笔记中使用的形式。43

This is the form Hilbert used in the published version of his November 20 note.43

• • •

• • •

我在这本书中给出了许多漂亮的方程,但广义相对论方程最为突出。我说“方程”,是因为这个单一的“方程”代表了十个方程(因为R μνT μν是对称的,所以有十个独立分量,就像我们在g μν中看到的一样)。然而,这几个优雅的符号是解开宇宙众多奥秘的钥匙——从水星近日点进动到大爆炸奇点。从光的弯曲(最终在 1919 年和 1922 年的日食探险中得到证实,以及 2019 年首次直接拍摄到的黑洞阴影的壮观图像)到 2015 年首次探测到的引力波。从引力对时钟和 GPS 的影响,到“引力磁力”、“参考系拖拽”、黑洞和引力透镜等更为深奥的事物——到目前为止,在每次测试中,这些方程都坚定不移。44我写完这本书之前,这种情况可能会改变——观测技术一直在进步!众所周知,我们还没有完整的量子引力理论,所以最终会出现一种新的理论。但它肯定必须在目前的精确度水平上与广义相对论相融合,就像爱因斯坦的理论与牛顿的理论相融合一样。

I’ve given a lot of beautiful equations in this book, but the equations of general relativity are standouts. I say “equations,” because this single “equation” represents ten equations (because Rμν and Tμν are symmetric, so there are ten independent components, just as we saw for gμν). Yet this elegant handful of symbols is the key that has unlocked so many mysteries of the universe—from the precession of Mercury’s perihelion to the Big Bang singularity. From the bending of light—which was eventually confirmed during the eclipse expeditions of 1919 and 1922, and by that spectacular first direct image of the shadow of a black hole in 2019—to the gravitational waves that were first detected in 2015. From the effect of gravity on clocks and GPS to more esoteric things such as “gravitomagnetism,” “frame-dragging,” black holes, and gravitational lenses—and so far, in every test the equations have stood firm.44 This might change before I finish writing this book—observational technology is improving all the time! And it is well known that we don’t yet have a complete theory of quantum gravity, so a new theory will eventually evolve. But it will surely have to mesh with general relativity at the current level of accuracy, just as Einstein’s theory does with Newton’s.

爱因斯坦没有亲眼见证他的理论的大部分惊人验证,但他知道自己发现了一些特别的东西——他成功的水星近日点和光线弯曲计算就是证明。多年后回首往事,他深刻地回忆起为此付出的巨大努力:

Einstein did not live to see most of these spectacular confirmations of his theory, but he knew he’d found something special—his successful Mercury perihelion and light bending calculations were proof of that. Looking back years later, he poignantly recalled the monumental effort it had taken:

在获得知识的光芒下,这种快乐的成就似乎是理所当然的,任何聪明的学生都可以毫不费力地掌握它。但在黑暗中探索的岁月,伴随着强烈的渴望,交替的信心和疲惫,以及最终的光明——只有经历过的人才能理解这一点。45

In the light of knowledge attained, the happy achievement seems almost a matter of course, and any intelligent student can grasp it without too much trouble. But the years searching in the dark, with their intense longing, their alternations of confidence and exhaustion and the final emergence into the light—only those who have experienced it can understand that.45

• • •

• • •

在 1916 年新理论概述的第一页,爱因斯坦对那些使广义相对论成为可能的数学家们表示了敬意:闵可夫斯基提出了时空概念;里奇和列维-奇维塔提出了张量微积分;高斯、黎曼和克里斯托费尔在非欧几里得几何方面做出了贡献,里奇对此进行了推广;当然,还有忠实的格罗斯曼,他不仅追踪并破译了相关论文,而且“还帮助我寻找引力场方程” 。46

On the first page of his 1916 overview of his new theory, Einstein paid tribute to the mathematicians whose work had made general relativity possible: Minkowski for formulating the idea of space-time; Ricci and Levi-Civita for tensor calculus; Gauss, Riemann, and Christoffel for their work in non-Euclidean geometry, which Ricci had generalised; and, of course, faithful Grossmann, who not only tracked down and deciphered the relevant papers but who “also helped me in my search for the field equations of gravitation.”46

1919 年的日食观测证实了爱因斯坦关于光弯曲的预测,广义相对论也使爱因斯坦一跃成为超级明星。观测结果公布后,全球各大媒体纷纷报道。1919年 11 月 7 日,伦敦《泰晤士报》报道称:“科学革命,宇宙新理论:牛顿思想被推翻。” 11 月 10 日,《纽约时报》刊登文章“天空中所有灯光都歪斜,科学家们对日食观测结果或多或少感到兴奋,爱因斯坦的理论胜利了”,诗意地暗指恒星弯曲的光路会改变它们在天空中的视位置。”

General relativity made Einstein a superstar, after his light-bending prediction was confirmed in the 1919 eclipse expedition—for when the expedition’s results were released, headlines trumpeted them around the world. “revolution in science, New Theory of the Universe: Newtonian Ideas Overthrown,” proclaimed the London Times on November 7, 1919. And on November 10, in a poetic allusion to the fact that the bending light path from the stars shifts their apparent positions in the sky, the New York Times ran with “lights all askew in the heavens, Men of Science More or Less Agog over Results of Eclipse Observations. einstein’s theory triumphs.”

爱因斯坦的成功也悄悄地使里奇名垂青史——至少在数学物理界是如此。贝尔特拉米等批评家曾抱怨说,与现有方法相比,张量分析没有任何实际好处,但爱因斯坦和格罗斯曼改变了这一切。正如里奇的好友列维-奇维塔所说,张量在广义相对论中的重要作用“也成为里奇‘荣耀的正当分配者’”,他的“伟大贡献”最终得到了官方认可。47

Einstein’s success also quietly immortalised Ricci—in the mathematical physics community, at least. Critics such as Beltrami had complained that tensor analysis had no practical benefit compared with existing methods, but Einstein and Grossmann had changed all that. As Ricci’s dear friend Levi-Civita put it, the essential role of tensors in general relativity “also became for Ricci ‘the just dispenser of glories,’” and his “great contribution” was officially recognised at last.47

(13)接下来发生了什么

(13) WHAT HAPPENED NEXT

正如我们在爱因斯坦关于下落的人和加速电梯的思想实验中看到的那样,在寻找自然规律的过程中,直觉的运用程度令人惊讶。单靠逻辑是远远不够的。然而,即使一个理论的方程式运行得非常好,也常常需要其他人的帮助来完善数学基础,或者更深入地阐明该理论的假设。这就是大卫·希尔伯特和菲利克斯·克莱因需要埃米·诺特的原因。

As we saw with Einstein’s thought experiment about falling people and accelerating elevators, a surprising amount of intuition goes into finding a law about how nature works. Logic alone is simply not enough. Yet even when a theory’s equations work spectacularly well, it is often necessary for others to come along and tighten up the mathematical foundations, or to shine a light more closely on the theory’s assumptions. Which is why David Hilbert and Felix Klein needed Emmy Noether.

诺特于 1915 年来到哥廷根。1907 年,她在埃尔朗根获得了博士学位,克莱因曾在那里任教,她是不变理论方面的专家——因此克莱因和希尔伯特邀请她来哥廷根,希望她能帮助他们解决有关广义相对论和能量的问题。她确实帮助了他们,因为 1917 年克莱因告诉希尔伯特:“你知道诺特小姐一直在为我的工作提供建议,多亏了她,我才弄清了这些问题。” 1

Noether arrived in Göttingen in 1915. She’d earned her doctorate in 1907 at Erlangen, where Klein had taught earlier, and she was an expert on invariant theory—so Klein and Hilbert had invited her to Göttingen, hoping she could help sort out their questions about general relativity and energy. And help them she did, for in 1917 Klein told Hilbert, “You know that Miss Noether advises me continually regarding my work, and it is only thanks to her that I have understood these questions.”1

尽管爱因斯坦有着传奇般的直觉,但他在尝试解释数学协方差(即如果你写一个张量方程,当你变换到另一组坐标时它将保持相同的形式)与物理学原理之间的关系时却遇到了困难。相对论和等价性引导他提出了他的理论。他的努力帮助后世的数学家们明确了坐标、参考系和点或事件变换之间的区别,以及作为不变性理论和对称群而非协变性理论的相对论之间的区别——但如果你对这些细微差别感到困惑,别担心:这些区别至今仍在争论中。2真正重要的是,爱因斯坦的张量方程对引力进行了极其精确的描述,由此人们正确地预测了许多非凡的后果。所以,我想在这里关注的是诺特在爱因斯坦于 1915 年底发表他的理论后立即发生的争论中帮助解决的问题,以及他 1916 年的更长的概述《广义相对论的基础》

Despite Einstein’s legendary intuition, he’d struggled when he tried to interpret the relationship between mathematical covariance—the idea that if you write a tensor equation, it will keep the same form when you transform to another set of coordinates—and the physical principles of relativity and equivalence that had guided him to his theory. His struggle helped later mathematicians sharpen the distinction between transformations of coordinates, frames of reference, and points or events, and between relativity as a theory of invariance and symmetry groups as opposed to one of covariance—but if you’re confused by such subtleties, don’t worry: these distinctions are still being debated today.2 All that matters, really, is that Einstein’s tensor equations give a marvelously accurate description of gravity, from which so many extraordinary consequences have been correctly predicted. So, what I want to focus on here are the issues that Noether helped resolve, during the debates that took place immediately after Einstein published his theory at the end of 1915, and his longer 1916 overview, The Foundation of the General Theory of Relativity.

顺便说一句,这篇 1916 年的论文是里奇张量微积分的大师级作品。由于这种语言对大多数物理学家和数学家来说仍然很陌生,爱因斯坦非常谨慎地简洁而清晰地阐述了我们在第 11 章中看到的张量规则。

By the way, this 1916 paper is a master class in Ricci’s tensor calculus. Because this language was still new to most physicists and mathematicians, Einstein took great care in setting out, economically but clearly, the tensor rules we saw in chapter 11.

1915 年 11 月,爱因斯坦终于得出了能量动量守恒方程T μν = 0,正如我们在上一章中看到的那样。他通过一条与希尔伯特不同的迂回路线得出了这一结论。1916 年 5 月,他问希尔伯特是否认为他们两种不同的方法背后可能存在某种深层原理。希尔伯特回答说,他认为可能有,而且他已经请“诺特小姐”来研究这个问题了。希尔伯特、克莱因和爱因斯坦等同事都对诺特怀有最高的敬意,但她的存在是如此独特,他们忍不住称她为“小姐”,而不是“博士”。也许他们认为这比直接叫她的姓氏更礼貌,因为他们之间经常这样称呼她。

In November 1915 Einstein had finally ended up with the conservation of energy-momentum equation T μν = 0, as we saw in the previous chapter. He’d come at it via a circuitous route that was different from Hilbert’s, and in May 1916 he asked if Hilbert thought there might be some deep principle underlying their two separate approaches. Hilbert replied that he thought there probably was, and that he’d already asked “Miss Noether” to investigate the issue. Hilbert and Klein, and colleagues such as Einstein, had the highest respect for Noether, yet her presence was so singular they couldn’t help calling her “Miss” rather than “Dr.” Perhaps they thought it was more polite than just using her surname, as they often did among themselves.

克莱因和希尔伯特试图解决的问题涉及广义相对论中能量动量守恒定律的物理和数学意义。普通力学中守恒定律的传统方法是变分法。这涉及到最小化“作用量”——即“拉格朗日量” L的积分,它是位置和速度的函数。莱昂哈德·欧拉和约瑟夫-路易斯·拉格朗日率先采用了这种方法,然后我们的矢量先驱威廉·罗文·汉密尔顿提出了一种有用的替代方法,其中L用动量而不是速度来表示(因此它是一个新函数,现在称为汉密尔顿函数,用H表示)。有时用动量而不是速度来表示更容易,但无论哪种方式,当LH用动能和势能来表示时,积分最小化时出现的运动方程有一个解,可以给出通常的能量守恒定律。这种方法的细节在这里并不重要:对于单个粒子,以这种方式找到的运动方程等​​同于牛顿第二运动定律,我们很快就会看到这如何导致守恒定律。

The questions Klein and Hilbert were trying to resolve concerned the physical and mathematical meaning of the energy-momentum conservation law in general relativity. The traditional route to conservation equations in ordinary mechanics had been the calculus of variations. This involves minimising the “action”—that is, the integral of a “Lagrangian” L, which is a function of position and velocity. Leonhard Euler and Joseph-Louis Lagrange had pioneered this method, and then our vector pioneer William Rowan Hamilton introduced a useful alternative, in which L is expressed in terms of momentum rather than velocity (so it is a new function, now called the Hamiltonian, denoted by H). Sometimes it is easier to work in terms of momentum rather than velocity, but either way, when L and H are expressed in terms of kinetic and potential energy, the equation of motion arising when the integral is minimised has a solution that gives the usual conservation of energy equation. The details of this method aren’t important here: for an individual particle the equation of motion found in this way is equivalent to Newton’s second law of motion, and we’ll see shortly how this leads to conservation equations.

与此同时,为了找到能量动量守恒方程,希尔伯特采用了优雅的拉格朗日方法,而爱因斯坦则采用了张量和哈密顿量的混合方法。他们开辟了新天地,因为他们首先必须弄清楚如何定义引力能——以及如何选择正确的LH来表达这种新能量。完成这些之后,仍然存在一个挥之不去的问题:结果方程T μν = 0 如何与传统的能量动量守恒思想相一致?

Meantime, to find their conservation of energy-momentum equations, Hilbert had used an elegant Lagrangian approach while Einstein had used a mix of tensors and a Hamiltonian. They were breaking new ground, because first they had to figure out how to define gravitational energy—and how to choose the right L or H to express this new kind of energy. When that was done, there was still a nagging question: How did the resulting equation, T μν = 0, gel with the traditional idea of energy-momentum conservation?

埃米·诺瑟和能量动量守恒

EMMY NOETHER AND THE CONSERVATION OF ENERGY-MOMENTUM

1916 年至 1917 年间,爱因斯坦、希尔伯特、克莱因和诺特就这个全新的引力能量问题交换了意见。每个人都参与了辩论,还有其他几个人加入进来,尤其是克莱因和希尔伯特以前的学生赫尔曼·外尔。但最终是诺特将所有这些联系在一起,她在 1918 年的论文《不变变分问题》中证明了著名的“诺特定理”。3这两个定理将守恒定律与“对称性”联系起来。而正如我们所见,对称性与“不变性”有关——不变性是指当你改变参考系时,某些事物不会改变。我们在图 9.19.3中也看到了这种不变性,当然它是张量方程的决定性特征。由于不变性与事物保持不变有关,因此它也与守恒有关——因为如果物理量不变,它就是“守恒的”。

Through 1916 and 1917, Einstein, Hilbert, Klein, and Noether exchanged ideas on this brand-new problem of gravitational energy. Each added to the debate, and several others joined in—notably Klein and Hilbert’s former student Hermann Weyl. But it was Noether who tied it all together, in the famous “Noether theorems” proved in her 1918 paper “Invariante Variationsprobleme.”3 These two theorems relate conservation laws to “symmetries.” And symmetries, as we’ve seen, relate to “invariance”—the idea that certain things are unchanged when you change your frame of reference. We also saw this kind of invariance in figures 9.1 and 9.3, and of course it is the defining feature of tensor equations. Since invariance has to do with things remaining the same, it also has to do with conservation—for if a physical quantity doesn’t change, it is “conserved.”

在普通力学(研究物体在力的作用下如何运动的学科)中,这种联系自拉格朗日以来就已为人所知。例如,我们已经知道引力F可以用势能V来表示:F = ∇ V 。但由于牛顿第二定律指出,作用在物体上的力等于物体动量矢量p的变化率,因此,我们有:

In ordinary mechanics—the study of the way objects move under forces—this connection had been known since Lagrange. For example, we’ve seen that gravitational force F can be expressed in terms of a potential V: F = ∇V. But since Newton’s second law says that the force acting on an object equals the rate of change of the object’s momentum vector, p, we have,

Fdd=++

Fdpdt=VVxi+Vyj+Vzk.

就分量而言,该方程给出d=, yz分量也是类似。现在讨论不变性,我们考虑一组坐标变换,它们在x方向上平移不同的量。如果势在这些变换下不变,那么它就不会改变——它在点 ( x, y, z )处的值与在 ( x + a, y, z ) 处的值相同:

In terms of components, this equation gives pxdt=Vx, and similarly for the y and z components. Now for the invariance, and let’s consider the group of coordinate transformations that are translations by various amounts in the x-direction. If the potential is invariant under these transformations, then it doesn’t change—it has the same value at the point (x, y, z) as it does at (x + a, y, z):

对于a的每一个值, V ( x + a, y, z ) = V ( x, y, z ) 。

V(x + a, y, z) = V(x, y, z), for every value of a.

由于使用x的哪个值并不重要,因此V必须与x无关。这意味着,这又意味着

Since it doesn’t matter which value of x is used, V must be independent of x. Which means Vx, and this, in turn, means that

d==,其中 C 是积分常数。

pxdt=Vxpx=C, where C is the constant of integration.

这意味着动量的x分量是恒定的,即守恒。

And this means that the x-component of the momentum is constant—it is conserved.

正如我之前提到的,使用拉格朗日方法或哈密顿方法可以得到相同的运动方程和相同的“守恒定律”。对于这种简单情况,这就像用大锤砸核桃一样,但对于包含许多粒子的复杂情况,这些方法通常更容易。

The same equations of motion, and the same “conservation” laws, are found using the Lagrangian or Hamiltonian methods, as I mentioned. This would be like using a sledgehammer to crack a walnut for this simple case, but these methods are often easier for complicated situations with many particles.

在诺特定理的最简单层面上,她证明了这是一个普遍的结果:当一个函数独立于一个特定的变量时,它表示某物是守恒的。例如,在广义相对论中,我们在图 12.1中看到,自由落体沿着类似于欧几里得几何直线的测地线运动,因此运动方程是关于测地线的方程——我们在第 12 章中看到,测地线取决于描述弯曲时空的度量。我们还看到,爱因斯坦选择度量的分量来扮演势的角色,类似于V。因此,就像我们刚刚看到的牛顿例子一样,如果存在一个框架,其中度量张量g μν独立于特定的时空坐标,那么动量的该分量沿粒子的路径守恒。然而,在时空中,“动量”是一个 4-D 矢量:它的空间分量是普通的空间动量分量,但时间分量被定义为能量。4

At the simplest level of her theorems, Noether showed that this is a general result: when a function is independent of a particular variable, it indicates that something is conserved. In general relativity, for example, we saw in figure 12.1 that free-falling objects move along geodesics analogous to the straight lines of Euclidean geometry, so the equation of motion is an equation about geodesics—and we saw in chapter 12 that geodesics depend on the metric that describes the curved space-time. We also saw that Einstein chose the components of the metric to play the role of the potential, analogous to V. So, like the Newtonian example we just saw, if there is a frame in which the metric tensor gμν is independent of a particular space-time coordinate, then that component of momentum is conserved along the particle’s path. In space-time, though, “momentum” is a 4-D vector: its spatial components are the ordinary spatial momentum components, but the time component is defined to be the energy.4

我在这里谈论的是单个运动物体的能量和动量,而不是用来描述引力场所携带的能量和动量的能量动量张量——但将粒子动量的时间分量标记为“能量”类似于我们在第 12 章中看到的时间分量T 41T 42T 43T 44。因此,有两个守恒定律问题需要解决:引力场中运动物体的能量和动量守恒定律——正如我们刚刚看到的,这与势Vg μν的对称性(或不变性或坐标独立性)有关——以及引力场本身的能量动量守恒定律,我们已经看到T μν ;ν = 0。诺特是第一个从数学上表明我们实际上是在谈论不同类型的守恒定律的人。

I’ve been talking about the energy and momentum of an individual moving object here, not the energy-momentum tensor used to describe the energy and momentum carried by a gravitational field—but labeling the time component of the particle’s momentum as “energy” is analogous to what we saw for the time components T41, T42, T43, T44 in chapter 12. So, there are two conservation issues to address: conservation of a moving object’s energy and momentum in a gravitational field—as we’ve just seen, this is related to the symmetry (or invariance or coordinate-independence) of the potentials, V and gμν—and conservation of the energy-momentum of the gravitational field itself, which we’ve seen is T μν = 0. Noether was the first to show mathematically that we’re actually talking about different types of conservation law.

我在尾注中概述了诺特如何做到这一点的本质。5结果是,在经典力学狭义相对论中,由此得出的守恒定律显然是物理的——它们源于牛顿定律所描述的物理条件。它们还涉及发散,正如我们在第 12 章中看到的电磁学一样。6方程T μν = 0 似乎涉及发散,但正如诺特证明的那样,它不是物理发散,而是数学类比

I’ve sketched the essence of how Noether did it in the endnote.5 The upshot is that in classical mechanics and special relativity, the resulting conservation laws are clearly physical—they arise from physical conditions such as those described by Newton’s laws. They also involve divergences, as we saw for electromagnetism in chapter 12.6 The equation T μν = 0 seems to involve a divergence, but as Noether proved, it is not a physical divergence but a mathematical analogy.

这并不是说T μν在广义相对论中不符合物理规律。确实,定义局部能量密度存在问题,除非存在某些时空中的对称性,没有全局守恒定律。然而,观察者可以定义某一点的能量密度,系统的总引力能量可以定义,引力波携带的能量通量也可以定义。7 因此,T μν宇宙学和引力辐射研究中起着至关重要的作用。而且,无论你想怎么称呼它,方程T μν = 0 都是寻找能量物质源与时空曲率之间关系所必需的。8

This is not to say that Tμν is unphysical in general relativity. True, there are problems in defining local energy density, and unless there are certain symmetries in the space-time, there’s no global conservation law. Still, an observer can define energy density at a point, and the total gravitational energy of a system can be defined, and so can the energy flux carried away by gravitational waves.7 So Tμν plays a vital role in the study of cosmology and gravitational radiation, for example. And, whatever you want to call it, the equation T μν = 0 is needed to find the relationship between energymatter sources and the curvature of space-time.8

图像

埃米·诺特 (Emmy Noether),约 1900 年。摄影师不详。Wikimedia Commons,公共领域。

Emmy Noether, circa 1900. Photographer unknown. Wikimedia Commons, public domain.

有了张量,一切都变得简单多了

IT’S A LOT SIMPLER WITH TENSORS

我们在第 12 章中看到,引力场方程是:

We saw in chapter 12 that the gravitational field equations are:

Rμ12μR=电视μ

Rμv12gμvR=KTμv.

我们在第 11 章中了解了协变导数,以及可以提高或降低张量指标而不改变其基本内容的事实9所以我要用楼上的指标写出引力场方程,然后对两边取协变导数:

And we met, in chapter 11, covariant derivatives, and the fact that you can raise and lower tensor indices without changing their essential content9so I’m going to write the gravitational field equations with upstairs indices, and then take the covariant derivative of both sides:

Rμ12μR=电视μ

Rμv12gμvR;v=kTμv;v.

现在,爱因斯坦和希尔伯特都发现,能量动量守恒意味着等式的右边为零。这意味着左边也必须为零。爱因斯坦和希尔伯特,克莱因和诺特都不知道,左边完全独立于守恒方程为零。这是因为所谓的“收缩比安奇恒等式”:

Now, both Einstein and Hilbert found that conservation of energymomentum means the right-hand side of the equation is zero. Which means the left-hand side must be zero too. What neither Einstein nor Hilbert knew, nor Klein nor Noether, was that the left-hand side is zero quite independently of the conservation equation. That’s because of what are known as “the contracted Bianchi identities”:

Rμ12μR=0.

Rμv12gμvR;v=0.

(之所以用复数是因为这个“方程”实际上是四个方程,每个分量 μ 一个。重复的指标 ν 表示和。)数学“恒等式”是一个始终为真的方程——因为根据定义,它必须为真。完整的比安基恒等式直接来自黎曼张量的定义,里奇在 19 世纪 80 年代就已经知道这一点,尽管他的宿敌路易吉·比安基在 1902 年重新发现了这些恒等式,首次发表了这一结果。(比安基赢得了里奇在 19 世纪 90 年代参加的意大利皇家数学奖,当里奇在 1901 年再次参加该奖时,比安基是一位不太同情的评委。)正如我们在第 12 章中看到的,里奇张量是由收缩黎曼张量形成的,所以这就是“收缩”比安基恒等式产生这个方程的方式。10

(The plural is because this “equation” is really four equations, one for each component μ. The repeated index ν indicates a sum.) A mathematical “identity” is an equation that is always true—because it must be true, by definition. The full Bianchi identities follow directly from the definition of the Riemann tensor, as Ricci had known in the 1880s, although it was his nemesis Luigi Bianchi who first published the result, when he rediscovered the identities in 1902. (Bianchi won the Italian Royal Mathematics Prize that Ricci had entered in the 1890s, and Bianchi was a none-toosympathetic judge when Ricci entered again in 1901.) As we saw in chapter 12, the Ricci tensor is formed from the contracted Riemann tensor, so that’s how the “contracted” Bianchi identities give rise to this equation.10

诺特阐明了守恒定律和对称性之间的关系,并详细解释了为什么T μν = 0 只是传统守恒定律的数学类比——但希尔伯特和克莱因假设这个守恒定律迫使Rμ12μR=0是正确的。因此,他们认为后一个方程是变分(拉格朗日和哈密顿)方法的结果,这种方法使爱因斯坦和希尔伯特得出了守恒定律方程。事实上,正如图利奥·列维-奇维塔在 1917 年指出的那样,反过来论证要简单得多,因为守恒定律方程通过张量比安奇恒等式直接由场方程导出!11

Noether had spelled out the relationship between conservation and symmetries, and had shown in detail why T μν = 0 is just a mathematical analogy to traditional conservation laws—but Hilbert and Klein had assumed that this conservation equation forced Rμv12gμvR;v=0 to be true. So, they regarded this latter equation as a consequence of the variational (Lagrangian and Hamiltonian) methods that had led Einstein and Hilbert to the conservation equation. In fact, as Tullio Levi-Civita noted in 1917, it’s much simpler to argue the other way, for the conservation equations follow directly from the field equations, via the tensorial Bianchi identities!11

顺便说一句,我在第 12 章中提到,爱因斯坦首先尝试用R μν = kT μν作为他的场方程。如果对方程两边求导,会得到 R μν = kT μν ,爱因斯坦理所当然地拒绝了这一公式,认为它不符合物理规律。12但是,比安奇恒等式立即表明,如果能量动量守恒定律成立,则这是一个错误的方程,因此T μν = 0。如果爱因斯坦知道这些恒等式,他就可以省去不少麻烦!

By the way, I mentioned in chapter 12 that Einstein had first tried Rμν = kTμν for his field equation. If you differentiate both sides of this equation, you get Rμν = kTμν, and Einstein rightly rejected this as unphysical.12 But the Bianchi identities show immediately that this is the wrong equation if conservation of energy-momentum is to hold, so that T μν = 0. Had Einstein known these identities, he’d have saved himself from quite a few headaches!

• • •

• • •

当然,诺特定理比我在这里指出的要复杂得多。几十年来,它们一直被忽视,因为除了数学复杂性之外,它们的意义还在于它们概括了大量已知的守恒定律。这些已知结果是几个世纪以来慢慢独立建立起来的,物理学家们花了一些时间才意识到诺特发现的这一重要性,即一个基本原理将数学对称性和物理守恒定律统一起来。但这一原理的普遍性也意味着她的定理比广义相对论适用范围更广:近年来,它们已应用于从量子力学到弹性力学到流体力学等,以及纯数学和数值分析。然而,早在 1918 年,甚至与广义相对论的联系也尚未完全理解。

Noether’s theorems are far more complicated than I’ve indicated here, of course. They were overlooked for many decades, because aside from their mathematical complexity, their significance lies in their generalising a host of already-known conservation results. These known results had been built up slowly and separately over the centuries, and it took physicists some time to appreciate the importance of Noether’s discovery that a single fundamental principle united mathematical symmetries and physical conservation laws. But the generality of this principle also means that her theorems are more widely applicable than general relativity: in recent times they have found applications from quantum mechanics to elasticity to fluid mechanics, for example, and in pure mathematics and numerical analysis. Back in 1918, though, even the link with general relativity was not fully understood.

事实上,直到 1924 年,Jan Schouten 和 Dirk Struik 才强调了诺特定性的拉格朗日恒等式与美丽的张量恒等式之间的联系Rμ12μR=0诺特利用了拉格朗日量的对称性,而比安奇恒等式则依赖于黎曼张量的对称性(当其指标的各种组合互换时,它保持不变,就像g μνT μν交换指标的顺序时保持不变一样:g μν = g νμT μν = T νμ。13

In fact, it wasn’t until 1924 that Jan Schouten and Dirk Struik highlighted the connection between Noether’s Lagrangian identities and the beautiful tensor identities Rμv12gμvR;v=0. Noether used the symmetries of the Lagrangian, while the Bianchi identities depend on the symmetries of the Riemann tensor (which remains invariant when various combinations of its indices are interchanged, just as gμν and Tμν are invariant when you swap the order of their indices: gμν = gνμ and Tμν = Tνμ).13

斯特鲁克很早就对相对论产生了兴趣,因为他在莱顿读书时,爱因斯坦曾在那里做过客座演讲——斯特鲁克的教授是爱因斯坦的密友保罗·埃伦费斯特,他曾在哥廷根与克莱因共事。半个世纪后,斯特鲁克仍然记得当时的兴奋之情埃伦费斯特的讲座让科学感觉“鲜活”——这是一项激动人心的当代活动,源自著名科学家和数学家之间的“冲突和辩论”,斯特鲁克回忆说,这些科学家和数学家包括克莱因、亚伯拉罕和爱因斯坦。14

Struik’s interest in relativity had arisen early, for while he was a student in Leiden, Einstein had given a guest lecture there—Struik’s professor was Einstein’s close friend Paul Ehrenfest, who’d worked with Klein at Göttingen. Half a century later Struik still remembered the excitement of Ehrenfest’s lectures, where science felt “alive”—an exciting, contemporary activity emerging “from conflict and debate” between famous living scientists and mathematicians, including, Struik recalled, Klein, Abraham, and Einstein.14

如果斯特鲁克的“冲突与辩论”一词让你想起马克思主义辩证法,那你就对了:斯特鲁克后来成为了一名科学史学家,并于 1936 年与他人共同创办了马克思主义杂志《科学与社会》,该杂志至今仍在发行。他受到这样一种理念的激励:科学既塑造社会,又被社会所塑造,科学家对其工作既负有科学责任,也负有社会责任——这一理念最终进入了科学课堂和伦理委员会。

If Struik’s phrase “conflict and debate” brings to mind Marxist dialectics, you’d be right: Struik later became a historian of science, and in 1936 he cofounded the Marxist journal Science and Society, which is still going today. He was motivated by the idea that science both shapes and is shaped by society, and that scientists have a social as well as a scientific responsibility for their work—an idea that has finally found its way into science classes and ethics committees.

诺特为获得认可而做出的努力

NOETHER’S STRUGGLE FOR ACCEPTANCE

1924-25 年,斯特鲁克和他的新婚妻子、同样是数学家的鲁思·拉姆勒博士在哥廷根度过了一年。“你必须脸皮厚才能生存,”他回忆道,因为“哥廷根数学家以讽刺幽默而闻名。”爱因斯坦也会同意这一点:“哥廷根人有时给我的印象不是他们想帮助人们清楚地阐述某个东西,而是他们只想向我们这些物理学家展示他们比我们聪明多少。”斯特鲁克指出,诺特“害羞又笨拙,经常成为笑柄。”但这不仅仅是性别歧视,因为斯特鲁克补充说,“善良的埃里希·贝塞尔-哈根”也受到了同样的“幽默” 。15

In 1924–25, Struik and his new wife, Dr. Ruth Ramler, who was also a mathematician, spent a year at Göttingen. “You had to have a thick skin to survive,” he recalled, for “the Göttingen mathematicians were known for their sarcastic humor.” Einstein would have agreed: “The people of Göttingen sometimes strike me,” he’d declared, “not as if they want to help one formulate something clearly, but as if they want only to show us physicists how much brighter they are than we.” Struik noted that Noether, “who was shy and rather clumsy, was often the butt of some joke.” But it wasn’t just sexism, for Struik added that “the good-natured Erich Bessel-Hagen” was also treated to the same “humor.”15

然而,性别歧视是诺特难以在学术界立足的原因。1915 年,在克莱因和希尔伯特的支持下,她申请了私人教师或私人讲师的资格。我之前提到过,当爱因斯坦在学术阶梯上迈出第一步时,他提交了他的狭义相对论论文,却因为“难以理解”而被拒绝。但诺特是因为厌女症而被拒绝的。在哥廷根,数学是哲学系的一部分,所以大多数成员都不知道诺特的数学才华——他们只看到她是个女人,而女人被允许教书是不可想象的。“当我们的士兵回到大学,发现他们被期望拜在女人脚下学习?”希帕蒂娅本可以告诉他们一些关于这方面的事情!事实上,希尔伯特回应说,他不认为“候选人的性别”是个问题:大学“不是澡堂”,他反驳道。16

Sexism was, however, the reason Noether found it hard to forge an academic career. In 1915, with the support of Klein and Hilbert, she had applied for habilitation as a private tutor or Privatdozent. I mentioned earlier that when Einstein presented his special relativity paper in his own first step on the academic ladder, it was rejected as “incomprehensible.” But Noether was rejected because of misogyny. At Göttingen, mathematics was part of the philosophy faculty, so most of the members had no idea of Noether’s mathematical brilliance—they just saw that she was a woman, and that it was unthinkable that a woman be allowed to teach. “What will our soldiers think when they return to the University and find that they are expected to learn at the feet of a woman?” Hypatia could have told them a thing or two about that! As it was, Hilbert responded that he didn’t see that “the sex of the candidate” was an issue: the university is “not a bathhouse,” he retorted.16

1918 年 5 月,在研究了诺特关于不变量的论文后,爱因斯坦写信给希尔伯特,说她的方法的普遍性给他留下了深刻的印象,并补充说:“如果哥廷根的老将们被送回诺特小姐的学校,那也没什么坏处。她似乎真的很懂行!”同年晚些时候,他研究了诺特小姐刚刚发表的守恒定律,他对此印象深刻,于是写信给克莱因说:“我再次感到,拒绝她任教是一种极大的不公。”他表示,如果克莱因太忙,他可以亲自与相关部门联系。幸运的是,战争终于结束了,德国建立了一个新的、更民主的政府——1919 年 6 月,诺特终于被允许成为一名私人讲师。17

In May 1918, after Einstein had studied a paper of Noether’s on invariants, he wrote to Hilbert saying how impressed he was with the generality of her approach, adding, “It would have done the Old Guard at Göttingen no harm to be sent back to school under Miss Noether. She really seems to know her trade!” Later that year he studied her conservation theorems hot off the press, and he was so impressed he wrote to Klein, “I once again feel that refusing her the right to teach is a great injustice.” He offered to approach the relevant ministry himself if Klein was too busy. Fortunately, the war finally ended, and a new, more democratic government was established in Germany—and in June 1919, Noether was finally allowed to become a Privatdozent.17

她的任教论文是她 1918 年的守恒定律,但在她的一生中,她因后来在抽象代数方面的工作而闻名——她的研究如此前沿,以至于被称为“现代代数之母”。18她也是第一位在相对论中发挥作用的女性,并且激励了许多在她之后有数学天赋的年轻女性。我记得我第一次参加广义相对论国际会议,大约有四百名男性和十名女性,其中包括巴黎大学教授伊冯娜·肖凯-布鲁阿,她从 20 世纪 50 年代起证明了一些具有里程碑意义的广义相对论定理,可谓传奇人物。我们仍然是少数,但情况一直在改善,女性的影响力越来越大。例如,2022 年,澳大利亚国立大学的苏珊·斯科特教授因其在引力方面的工作,包括她在 2015 年探测引力波中所发挥的作用,获得了欧洲科学院的布莱斯·帕斯卡奖章。在相关话题上,凯蒂·布曼是哈佛大学的博士后研究员,她在设计导致 2019 年首次直接拍摄黑洞图像的算法方面发挥了关键作用。

Her habilitation thesis had been her 1918 conservation theorems, but in her lifetime, she was much more famous for her later work on abstract algebra—she was so cutting-edge she’s been dubbed “the mother of modern algebra.”18 But she was also the first woman to play a role in relativity theory, and she’s been an inspiration to many mathematically inclined young women who have come after her. I remember my first international conference on general relativity, where there were about four hundred men and ten women—including University of Paris professor Yvonne Choquet-Bruhat, who was a living legend for proving, from the 1950s on, some landmark theorems in general relativity. We are still a minority, but things are improving all the time, and women are increasingly visible. For example, in 2022, the Australian National University’s Professor Susan Scott won the European Academy of Sciences’ Blaise Pascal Medal for her work on gravitation, including her role in the 2015 detection of gravitational waves. And on a related topic, Katie Bouman was a Harvard postdoctoral fellow when she famously played a key role in devising the algorithms that led to the first direct image of a black hole in 2019.

黑洞和引力波是广义相对论的预测,尽管爱因斯坦本人对它们存在的可能性持矛盾态度。物理存在。但那不重要:一切都在那些非凡的小张量方程中,

Black holes and gravitational waves are predictions of general relativity, although Einstein himself was ambivalent about the possibility of their physical existence. But that doesn’t matter: it’s all there in those extraordinary little tensor equations,

Rμ12μR=电视μ

Rμv12gμvR=KTμv.

至少,对于那些知道如何解决这些方程的人来说,它是存在的,而如今这通常需要使用数值算法和计算机能力。诸如 Ricci 和度量张量之类的张量是它的核心,尽管不一定存在于数值方法本身中。但对于我们这些致力于精确解的人来说,这些张量指标及其对称性有助于减轻我们的计算任务负担。

At least, it’s there for those who know how to solve these equations, and today this often requires the use of numerical algorithms and computer power. Tensors such as the Ricci and metric tensors are there at the heart of it, although not necessarily in the numerical methods themselves. But for those of us working on exact solutions, those tensor indices and their symmetry properties help lighten the load of our computational tasks.

“平行”的含义

THE MEANING OF “PARALLEL”

广义相对论诞生后,还有另一个有趣的张量问题得到了解决。在图 3.1中,我们看到了向量加法的平行四边形规则,通过平移或“移动”向量A到B的末端将其加到 B 上同时保持A与其自身平行。或者,你可以移动B,使其保持自身平行。在球体这样的曲面上,你不能像这样移动向量:正如我所提到的,在赤道处平行的经线在两极不再平行。显然,通常的平行概念只有在曲线近似于直线时才在局部有意义。

There’s another interesting tensor problem that was solved in the wake of general relativity. In the parallelogram rule for vector addition, which we saw back in figure 3.1, vector A is added to B by translating or “transporting” it to align with the end of B, all the while keeping A parallel to itself. Alternatively, you can move B, keeping it parallel to itself. On a curved surface such as that of a globe, you cannot move vectors around like this: as I’ve mentioned, meridians of longitude that are parallel at the equator are no longer so at the poles. Evidently, then, the usual idea of parallelism makes sense only locally, when a curved line is approximately straight.

图像

图 13.1 。在平面空间中,平行移动一个从点A开始垂直的向量是没有问题的——但在曲面上,这个概念必须仔细定义。我在相关框中对此做了进一步解释。

FIGURE 13.1. Parallel transporting a vector that starts out being vertical at point A is unproblematic in flat space—but on a curved surface, the notion must be carefully defined. I’ve explained this a little more in the related box.

1917 年,列维-奇维塔 (Levi-Civita) 发明了如何沿曲面“平行传输”矢量,他用张量实现了这一目标。他的“平行传输”方程(在方框中)有助于显示曲率导致两条最初平行的测地线相对于彼此偏离的程度,就像球体上会聚的子午线一样。在黑洞周围等弯曲巨大的时空中,测地线会如此迅速而剧烈地汇聚,如果你靠得太近,就会被压碎。(平方反比定律表明,你也会被撕裂,因为重力会随着你头和脚之间的距离而发生巨大变化!)

It was Levi-Civita who figured out, in 1917, how to “parallel transport” vectors along curved surfaces—and he did it with tensors. His “parallel transport” equation (in the box) helps show how much the curvature causes two initially parallel geodesics to deviate with respect to each other—like the converging meridians on a globe. In a hugely curved space-time such as that around a black hole, the geodesics converge so quickly and drastically that you would be crushed if you came too close. (The inverse square law shows that you’d also be torn apart, because the gravity changes so dramatically over the distance between your head and your feet!)

平行传输作为表征曲率的一种方式

PARALLEL TRANSPORT AS A WAY OF CHARACTERISING CURVATURE

要了解列维-奇维塔的平行传输思想与曲率有何关系,最简单的方法是从平坦空间中的想法开始。想象一张平整的纸上画着一个三角形(如图 13.1所示)。在点A处垂直握住一支铅笔(代表一个矢量),然后保持铅笔与自身平行,沿着平面页面移动到点C。现在做同样的事情,但朝相反方向移动到C ,因此你将沿着从AB再到C的直线传输垂直的铅笔。这并不奇怪——铅笔始终保持垂直。如果你采用内在的蚂蚁视角,将矢量保持在页面的二维平面上,也会发生同样的事情:首先,假设有一个与边AC相切的矢量。当你将它从A移动到C时,无论朝哪个方向传输,它都将保持与自身平行。

To see how Levi-Civita’s idea of parallel transport relates to curvature, it’s easiest to start with the idea in flat space. Imagine a flat piece of paper with a triangle drawn on it (as in fig. 13.1). Hold a pencil—standing in for a vector—vertically at the point A, and then, keeping it parallel to itself, move it along the flat page to the point C. Now do the same thing, but travel to C in the opposite direction, so you’re transporting the vertical pencil along the line from A to B and then up to C. No surprises there—the pencil remains vertical all the time. The same thing happens if you take an intrinsic, ant’s-eye view, keeping your vector in the 2-D plane of the page: begin with, say, a vector tangent to the side AC. It will remain parallel to itself when you transport it from A to C in either direction.

现在想象一下将铅笔垂直放在球或橙子“赤道”上的A点。在这个垂直位置,铅笔与A处的球相切并指向上方。将其向上移向C处的极点,保持铅笔与球相切但保持铅笔笔直(确保不要扭曲它)。现在换个方向,平行移动铅笔沿着曲线从AB再到C 。这次铅笔(或矢量)最终与从A直接移动到C 的相同矢量成 90 度角。这就是非交换性的作用!

Now imagine holding the pencil vertically at a point A on the “equator” of a ball or orange. In this vertical position the pencil is tangent to the ball at A and pointing upward. Move it up toward the pole at C, keeping it tangent to the sphere but holding it straight (make sure you don’t twist it). Now go the other way, parallel transporting the pencil along the curve from A to B and then up to C. This time the pencil—or vector—ends up at an angle of 90 degrees from the same vector transported directly from A to C. This is noncommutativity in action!

汉密尔顿可能会惊讶,他的“令人震惊”的观点,即数学运算并不总是可交换的,竟然得到了如此多的应用——这次是作为曲率的度量。这是我们的二维蚂蚁外星人可以发现的固有曲率,只需用铅笔沿着表面爬行并注意它在C处如何改变方向即可。

Hamilton would be amazed that his “shocking” notion that mathematical operations are not always commutative has found so many applications—this time, as a measure of curvature. It is the kind of intrinsic curvature our 2-D ant-alien could discover, simply by crawling along the surface with a pencil and noting how it changed direction at C.

微积分

CALCULUS, TOO

在普通向量分析中,当你对向量进行微分时,直观地说,你会比较两个相邻点的向量,然后除以它们之间的距离——并且当你计算一个点和另一个点的值时,你会隐式地假设向量保持平行。因此,如果我们要理解如何定义弯曲空间中的导数(即里奇所说的“协变”导数),向量必须能够以某种平行的方式沿曲线移动或“传输”。因此,列维-奇维塔的平行传输思想将协变导数和曲率联系起来也就不足为奇了。

When you differentiate vectors in ordinary vector analysis, intuitively speaking you compare the vector at two nearby points and divide by the distance between them—and you implicitly assume the vector remains parallel when you calculate its value at one point and then the other. So, vectors must be able to be moved or “transported” along a curve in some sort of parallel way, if we are to understand how to define derivatives in curved spaces (that is, what Ricci called “covariant” derivatives). It’s not surprising, then, that Levi-Civita’s idea of parallel transport connects covariant derivatives and curvature.

原来向量V沿切向量U的曲线平行传输的定义是U μ V ν = 0。(分号表示协变导数。)

It turns out that the definition of the parallel transport of a vector V along a curve with tangent vector U is UμVν = 0. (The semicolon denotes the covariant derivative.)

在科学和数学领域,经常会发生这样的事情:大约在同一时间,其他人独立发现了与列维-奇维塔相同的思想——尽管他还没有发表这些思想。这个人就是扬·斯考滕。他的同事德克·斯特鲁克回忆起斯考滕冲进办公室挥舞着列维-奇维塔论文的那天。“他也有我的测地线运动系统,”斯考滕告诉斯特鲁克,“只是他称它们为平行。”斯特鲁克回忆说,列维-奇维塔的方法比斯考滕的简单得多;不过,他沉思道,“很少有人意识到,斯考滕差点就获得了自里奇发明张量微积分以来最重要的发现的荣誉。” 19

As so often happens in science and mathematics, someone else had independently discovered the same ideas as Levi-Civita, at around the same time—although he hadn’t yet published them. This someone was Jan Schouten. His colleague Dirk Struik recalled the day Schouten had come bursting into his office waving Levi-Civita’s paper. “He also has my geodesically moving systems,” Schouten told Struik, “only he calls them parallel.” Struik recalled that Levi-Civita’s approach was much simpler than Schouten’s; still, he mused, “Few people realize that Schouten barely missed getting credit for the most important discovery in tensor calculus since its invention by Ricci.”19

斯特鲁克于1923年与列维-奇维塔共事,称他为活泼、温和、迷人。颇具影响力的苏格兰代数几何学家威廉·霍奇写道,列维-奇维塔是“他那个时代最知名、最受欢迎的数学家之一”。20和斯特鲁克一样,列维-奇维塔也他娶了数学毕业生利比里亚·特雷维萨尼为妻,她是他以前的学生。她原本希望教数学,但最终和丈夫一起周游世界,丈夫在巡回演讲中很受欢迎。不幸的是,这一切都在 20 世纪 30 年代发生了变化,当时法西斯和纳粹开始迫害,列维-奇维塔、爱因斯坦、诺特和许多其他犹太学者被剥夺了学术职位,甚至失去了生命。

Struik worked with Levi-Civita in 1923 and described him as vivacious, gentle, and charming. The influential Scottish algebraic geometer William Hodge wrote that Levi-Civita was “one of the personally best known and best liked mathematicians of his time.”20 And like Struik, Levi-Civita had married a maths graduate, Liberia Trevisani—his former student. She had hoped to teach mathematics, but ended up traveling the world with her husband, who was in great demand on the lecture circuit. Tragically, all that would change in the 1930s, when the Fascists and Nazis began their persecution, and Levi-Civita, Einstein, Noether, and so many other Jewish academics were stripped of their academic positions, if not of their lives.

阿诺德·索末菲和大卫·希尔伯特等人对他们国家在两次世界大战之间的发展方向感到震惊。第一次世界大战结束后不久,1919 年日食探险队的队长亚瑟·爱丁顿曾看到英国团队证实德国理论(爱因斯坦的光弯曲预测)的象征意义,这给了他们希望。但这两个昔日敌对国家之间的敌意仍在酝酿,当意大利数学家于 1928 年组织第一次战后数学大会时,他们不遗余力地邀请德国同行参加。许多德国数学家拒绝参加,但当希尔伯特得意洋洋地带领他的同胞代表团参加开幕式时,他们受到了全场起立鼓掌。希尔伯特在会上发表的讲话肯定会让爱丁顿感到高兴。他说,数学没有界限,包括国家界限。“根据人和种族来构建差异是对我们科学的完全误解。对于数学来说,整个世界就是一个国家。” 21这是一种崇高的情操——从国际合作对数学的发展一直很重要的角度来看,数学现在是一种国际语言。但战争改变了一切。例如,在 2022 年,在弗拉基米尔·普京灾难性地入侵乌克兰之后,数学家之间也发生了类似的争论,因为一份期刊拒绝发表俄罗斯机构的论文,国际数学联盟剥夺了圣彼得堡举办 2022 年国际数学家大会的权利。一些人和希尔伯特一样认为不应该有歧视,而另一些人则认为制裁传达了机构个人对战争负责的信息(例如希尔伯特和爱因斯坦在第一次世界大战期间拒绝签署德皇的责任豁免书时就表明了这一点)。22

The likes of Arnold Sommerfeld and David Hilbert were appalled at the direction their country had taken between the wars. Soon after the first war, Arthur Eddington, a leader of the 1919 eclipse expedition, had seen hope in the symbolism of a British team confirming a German theory—Einstein’s light-bending prediction. But hostility between the former enemy nations still simmered, and when Italian mathematicians organised the first postwar mathematics congress in 1928, they took pains to invite their German colleagues. Many German mathematicians refused to go, but when Hilbert triumphantly led a delegation of his countrymen into the opening session, they received a standing ovation. Hilbert addressed the gathering in terms that would certainly have pleased Eddington. There are no limits in mathematics, he said, including national ones. “It is a complete misunderstanding of our science to construct differences according to people and races. For mathematics, the whole world is a single country.”21 It’s a noble sentiment—and it is true in the sense that international collaboration has always been important in the advancement of mathematics, which is now an international language. But war changes everything. In 2022, for example, in the wake of Vladimir Putin’s disastrous invasion of Ukraine, there was a similar debate among mathematicians, after a journal refused to publish papers from Russian institutions, and the International Mathematics Union stripped St. Petersburg of its right to host the 2022 International Congress of Mathematicians. Some felt as Hilbert did, that there should be no discrimination, and others felt that sanctions sent a message about institutional and individual responsibility for war (such as Hilbert and Einstein had shown when they refused to sign the kaiser’s waiver of responsibility during World War I).22

• • •

• • •

列维-奇维塔的不变张量平行定义对于张量数学非常重要——尤其是对于理解协变微分的概念。但在 1929 年,爱因斯坦开始与法国数学家埃利·嘉当 (Élie Cartan) 通信,探讨具有更广泛平行定义的空间的可能性。嘉当是将其前辈的所有思想——包括长期被忽视的赫尔曼·格拉斯曼 (Hermann Grassmann)——融入里奇张量微积分的现代“微分形式”版本中做出最大贡献的人,而格拉斯曼正是嘉当的研究对象。例如,嘉当阐明了单形式作为矢量对偶的概念,我在第 11 章中简要提到过这一点。

Levi-Civita’s invariant tensor definition of parallelism was important for the maths of tensors—especially for understanding the idea of covariant differentiation. But in 1929 Einstein began corresponding with the French mathematician Élie Cartan on the possibility of spaces with a broader definition of parallelism. Cartan is the one who did the most to put all the ideas of his predecessors—including the long-neglected Hermann Grassmann, whom Cartan had studied well—into the modern “differential form” version of Ricci’s tensor calculus. It was Cartan who, for example, clarified the idea of one-forms as duals of vectors, which I mentioned briefly in chapter 11.

嘉当和爱因斯坦之间的信件内容与嘉当的“绝对”平行性概念有关,这个概念允许“平行”矢量发生扭转或扭曲。正如我在图 13.1中用铅笔说明的那样,这在列维-奇维塔的平行传输定义中是不允许的,因此它不用于广义相对论(因为非扭转平行传输与爱因斯坦方程中的协变导数有关)。爱因斯坦曾希望增加扭转可以成为统一电磁学和引力的一种方法,但就像希尔伯特和古斯塔夫·米一样,他没有成功。尽管如此,具有扭转的时空(例如爱因斯坦-嘉当理论中的时空)至今仍在探索中,例如作为避免大爆炸奇点的可能方法,或用于解释物质的内在自旋。

The content of the letters between Cartan and Einstein has to do with Cartan’s idea of “absolute” parallelism, a notion that allowed torsion or twisting of “parallel” vectors. As I mentioned with the pencil illustration in figure 13.1, this isn’t allowed in Levi-Civita’s definition of parallel transport—and therefore it isn’t used in general relativity (because nontwisting parallel transport relates to the covariant derivatives in Einstein’s equations). Einstein had hoped adding torsion might be a way to unify electromagnetism and gravity—but like Hilbert and Gustav Mie, he didn’t succeed. Nonetheless, space-times with torsion, such as those in the Einstein-Cartan theory, are still being explored today—as a possible way of avoiding the Big Bang singularity, for example, or to account for the intrinsic spin of matter.

虽然爱因斯坦和嘉当之间的通信细节超出了我的范围,但粗略一看就会发现,他们使用的数学语言是张量。这也表明了两人的互动方式是相互尊重的。1929 年,50 岁的爱因斯坦写道:“我很幸运能有你这样的同事。因为你正好拥有我所缺乏的东西:令人羡慕的数学能力……你为这个问题付出了如此多的努力,我既感动又高兴。”60 岁的嘉当写道:“我很自豪我的信件可能会引起你的兴趣……我认为这是一种荣幸,”他补充道,“你愿意抽出一些时间给我,这对科学来说非常宝贵。” 23

Although the details of the Einstein-Cartan correspondence are beyond my scope here, a cursory glance shows that tensors were the mathematical language they were speaking. It also shows the respectful way the two men interacted. Einstein, who was fifty in 1929, wrote such things as, “I am very fortunate that I have acquired you as a coworker. For you have exactly that which I lack: an enviable facility in mathematics.... I am both touched and delighted that you have taken so many pains over the problem.” And sixty-year-old Cartan wrote, “I’m very proud my letters may be of some interest to you.... I consider it to be a privilege,” he added, “that you are willing to spare me some of your time, which is so precious for science.”23

爱因斯坦 1931 年 6 月 13 日的信尤其令人心酸。他告诉嘉当,他的老朋友马塞尔·格罗斯曼发表了一篇论文,对绝对并行性进行了“粗鲁”的批评,但他想让嘉当知道格罗斯曼身患晚期多发性硬化症,病情严重。爱因斯坦忠诚地说道:“我告诉你这些,是为了敦促你不要公开回应他。”他还补充说,格罗斯曼病情严重,无法为他批评中令人不快的语气和误导性的内容负责。五年后,格罗斯曼因长期患病而去世,爱因斯坦动情地写信给他的遗孀,告诉她马塞尔是一位多么珍贵的朋友。24

Einstein’s letter of June 13, 1931, is especially poignant. He informs Cartan that his old friend Marcel Grossmann has published a paper “rudely” criticising the idea of absolute parallelism, but he wants Cartan to know that Grossmann is seriously ill with advanced multiple sclerosis. “I tell you all this to urge you not to answer him publicly,” Einstein said loyally, adding that Grossmann was too ill to be accountable for the unpleasant tone and misguided content of his critique. When Grossmann died five years later, after his long and debilitating illness, Einstein wrote movingly to his widow, telling her what a treasured friend Marcel had been.24

格罗斯曼可能从未意识到自己在张量理论的普及过程中发挥了多么关键的作用。但自 1975 年以来,他在广义相对论方面的贡献在马塞尔·格罗斯曼会议中得到了表彰。每隔三四年,来自世界各地的研究人员就会齐聚一堂,讨论最新进展。在表彰格罗斯曼的同时,这些会议也至少暗中表彰了里奇和列维-奇维塔的数学才华以及爱因斯坦的天才。

Grossmann may never have realised what a crucial role he had played in putting tensors on the map. But since 1975 his legacy in general relativity has been honoured in the Marcel Grossmann Meetings, which, every three or four years, bring together researchers from all over the world to discuss the latest developments. And in honouring Grossmann, these meetings also honour—implicitly, at least—the mathematical brilliance of Ricci and Levi-Civita, and the genius of Einstein.

结语

EPILOGUE

广义相对论的成功——尤其是在 1919 年日食考察之后——将张量带入主流。例如,保罗·狄拉克在他的量子电动力学 (QED) 方程中使用了张量以及矢量和矩阵。在这个可以追溯到 1927 年的理论中,狄拉克将麦克斯韦的电磁理论、电子量子力学(用“狄拉克方程”描述)和狭义相对论结合起来。这是一项了不起的成就——而狄拉克当时只有 25 岁。

The success of general relativity—especially after the 1919 eclipse expedition—brought tensors into the mainstream. For example, Paul Dirac used them, along with vectors and matrices, in his equations of quantum electrodynamics (QED). In this theory, which dates from 1927, Dirac united Maxwell’s theory of electromagnetism, the quantum mechanics of the electron (described by “Dirac’s equation”), and special relativity. It was a spectacular achievement—and Dirac was only twenty-five years old.

狄拉克的 QED 帮助创立了量子场论(因为 QED 处理的是带电粒子和电磁场之间的相互作用)。这种量子理论是分析粒子对撞机中粒子所必需的,例如,因为这些粒子被加速到如此高的速度,以至于相对论效应非常显著。粒子加速器(如欧洲核子研究中心著名的大型强子对撞机)旨在寻找各种理论预测的新粒子——2012 年发现希格斯玻色子在当时是重大新闻。如今,一些物理学家怀疑这样做是否值得。1然而,早在 1929 年底,狄拉克就利用他的理论预测了反物质这一奇异现象的存在——特别是“正电子”的存在,正电子是带负电荷的电子的“镜像”。三年后,人们通过实验发现了正电子,当时使用的是云如今,它们已在医学成像过程中得到广泛应用,例如 PET(正电子发射断层扫描)等。2

Dirac’s QED helped launch quantum field theory (because QED deals with interactions between charged particles and electromagnetic fields). This is the kind of quantum theory that is needed to analyse the particles in particle colliders, for example, because these particles are accelerated to such high speeds that relativistic effects are significant. Particle accelerators such as CERN’s famous Large Hadron Collider aim to find new particles predicted by various theories—and the detection of the Higgs boson in 2012 was big news at the time. Nowadays, some physicists wonder if the cost has been worth it.1 Back at the end of 1929, however, Dirac used his theory to predict the existence of the bizarre phenomenon of antimatter—in particular, the existence of “positrons,” positively charged particles that are “mirror images” of electrons, which carry a negative charge. Three years later, positrons were discovered experimentally, using a cloud chamber, and today they are used routinely—for example, in medical imaging processes such as PET (positron emission tomography).2

狄拉克方程描述了单个电子的行为,但在 1936 年,杰出的数学物理学家、格顿学院毕业生 Bertha Swirles 将狄拉克的相对论量子分析扩展到两个电子,包括它们的电子自旋相互作用。3诺特和奇泽姆·杨一样,Swirles 也曾在哥廷根工作过,在那里她与量子理论的两位先驱马克斯·玻恩和维尔纳·海森堡一起工作。1940 年,她与剑桥地球物理学家哈罗德·杰弗里斯结婚,并共同编写了一本数学物理教科书。她还继续自己的量子理论研究,发表了许多论文,并在她以前的学院格顿学院担任数学讲师。

Dirac’s equation described the behavior of a single electron, but in 1936, the remarkable mathematical physicist Bertha Swirles, a Girton graduate, extended Dirac’s relativistic quantum analysis to two electrons, including their electron spin interactions.3 Like Noether and Chisholm Young, Swirles spent time at Göttingen, where she worked with Max Born and Werner Heisenberg, two of the pioneers of quantum theory. In 1940, she married Cambridge geophysicist Harold Jeffreys, with whom she wrote a textbook on mathematical physics. She also continued her own research in quantum theory, publishing many papers, and took up a mathematics lectureship at her former college, Girton.

如今,相对论量子力学(包括 QED)在许多其他领域都很重要,包括量子化学、材料科学和其他纳米技术应用。我们已经看到电子自旋有许多应用,从 MRI 到量子计算机中的量子比特。至于张量和量子力学,我们在第 11 章中看到,矢量和张量积对于确定粒子集合或量子比特的量子态非常重要。张量积还用于证明不可能复制量子比特(不可克隆定理),并用于分析诸如量子隐形传态和纠缠等诱人的事情。

Today relativistic quantum mechanics (including QED) is important in many other areas, including quantum chemistry, materials science, and other nanotech applications. And we’ve already seen that electron spin has many applications, from MRIs to the qubits in quantum computers. As for tensors and quantum mechanics, we saw in chapter 11 that vectors and tensor products are important in figuring out the quantum state of a collection of particles or qubits. Tensor products are also used to prove that it is impossible to copy a qubit (the no-cloning theorem), and in analysing such tantalising things as quantum teleportation and entanglement.

然后是相对论宇宙学,这是爱因斯坦在完成广义相对论后不久开创的。当然,张量是关键——尤其是度量,以及爱因斯坦方程中的其他张量。我们之前看到,爱因斯坦创造了一个静态时空,例如可以描述地球或静止恒星等物体周围的时空,这些物体的引力场不会随时间而变化。1917 年,他开始对宇宙本身做同样的事情。

Then there is relativistic cosmology, which Einstein pioneered soon after he’d finished his theory of general relativity. Tensors are key, of course— especially the metric, and the other tensors in Einstein’s equations. We saw earlier that Einstein had created a static space-time, such as might describe the space-time around an object such as Earth or a quiescent star, whose gravitational field doesn’t change in time. In 1917 he set out to do the same for the cosmos itself.

宇宙学的理念是做出一些关于宇宙中物质空间分布的假设——这些假设将给出必须反映在时空度量中的必要特征。爱因斯坦假设他的宇宙度量与时间无关,因为和当时几乎所有人一样,他认为宇宙是静态的。毕竟,太阳恒星继续燃烧,恒星也保持在天空中通常的位置和路径上——至少就天文学家所知而言。爱因斯坦还假设宇宙是球对称的,因为总体而言,恒星和星系看起来是对称分布的。但问题是引力会导致物质吸引其他物质——你会认为这意味着所有恒星和星系会不断地聚集在一起,最终合并。由于它们不会这样做,而且宇宙似乎是静态的,一切都保持在正确的位置,爱因斯坦在场方程中添加了一个新项。这个额外的项就是臭名昭著的“宇宙常数”,爱因斯坦添加它是为了抵消引力的自然吸引力。

The idea in cosmology is to make some assumptions about the spatial distribution of matter in the universe—assumptions that will give essential features that must be reflected in the space-time metric. Einstein assumed his cosmological metric was independent of time, for like nearly everyone else at the time, he believed the universe was static. After all, the Sun kept on burning and the stars kept to their usual positions and paths in the sky—as far as astronomers could tell, anyway. Einstein also assumed the universe was spherically symmetric, because, overall, the stars and galaxies appear to be symmetrically distributed. But the problem was that gravity causes matter to attract other matter—which you’d think would mean that all the stars and galaxies would be continually drawn together, ultimately coalescing. Since they don’t do this, and since the cosmos appeared to be static, with everything staying happily in its right place, Einstein added a new term to the field equations. This extra term is the infamous “cosmological constant,” which Einstein added to counteract gravity’s natural attractive tendency.

爱因斯坦的静态模型是第一个宇宙相对论模型。然而,令他恼火的是,20 世纪 20 年代,亚历山大·弗里德曼等数学家发现广义相对论预测宇宙应该在膨胀(或收缩)。直到 1929 年,当埃德温·哈勃找到宇宙确实在膨胀的确凿证据时,爱因斯坦才放弃了他对静态宇宙的信念——在这个过程中放弃了宇宙常数。他曾称这个常数是他最大的错误,但今天他已经得到了救赎:宇宙常数又回来了——至少是暂时的。由于它被设计为“推回”引力,它似乎是寻找神秘暗能量的理想工具,这种暗能量不仅导致宇宙膨胀,而且以加速的速度膨胀。

Einstein’s static model was the very first relativistic model of the cosmos. Much to his annoyance, though, in the 1920s mathematicians such as Alexander Friedmann found that general relativity predicted the universe should be expanding (or contracting). It was only in 1929, when Edwin Hubble found definitive evidence that the universe is indeed expanding, that Einstein abandoned his belief in a static cosmos—abandoning the cosmological constant in the process. He famously called this constant his biggest blunder, but today he has been redeemed: the cosmological constant is back—at least tentatively. Since it was designed to “push back” on gravity, it seems to be an ideal tool in the search for the mysterious dark energy causing the universe not just to expand, but to do so at an accelerating rate.

• • •

• • •

我可以继续说下去,但最后我必须停下来!可以说,在当今的数学物理中,向量和张量至关重要。而且它们仍在不断发展,正如我们在第 11 章中看到的张量的现代解释,以及嘉当对现代微分几何的发展。但向量和张量不仅在数学和物理学中有用。它们在工程和化学中也很重要——例如,从模拟涡轮机和飞机发动机叶片上的气流到晶体学。它们在机器学习、搜索等数字技术中也很重要引擎、计算机视觉和自然语言处理,正如我们在第 11 章中看到的,以及在第 9 章中看到的神经网络。

I could go on, but I must finally stop! It’s enough to say that in mathematical physics today, vectors and tensors are vital. And they are still evolving, as we saw with the modern interpretation of tensors in chapter 11, and with Cartan’s development of modern differential geometry. But it’s not just in mathematics and physics that vectors and tensors are useful. They’re important in engineering and chemistry, too—from modeling airflow over the blades in turbines and aircraft engines to crystallography, for example. They’re important in digital technologies such as machine learning, search engines, computer vision, and natural language processing, as we saw in chapter 11, and in the neural networks we saw in chapter 9.

张量在所有这些领域都很重要,主要有两个原因:一是用于解决涉及不变性的问题(例如在广义相对论、量子力学和神经网络中),二是用于表示和处理数据。当古代美索不达米亚数学家将数据表刻在泥板上时,他们肯定没有想到当今数据科学中张量需要处理的数据量如此之大。当古代中国数学家使用矩阵状数组求解线性方程组时,他们也不知道这些系统如今有多么复杂和多样——涵盖了从优化业务成本和游戏策略到机器人技术和人工智能,再到我们在第 4 章中看到的 Google 网页排名算法等所有内容

Tensors are important in all these areas for two main reasons: for solving problems involving invariance—in general relativity, quantum mechanics, and neural networks, for example—and for representing and handling data. When ancient Mesopotamian mathematicians etched tables of data into clay tablets, they surely never imagined the huge amount of data that tensors need to handle in data science today. And when ancient Chinese mathematicians used matrix-like arrays to solve systems of linear equations, they would have had no idea how complex and diverse these systems are today—covering everything from optimising business costs and game strategies to robotics and AI to the Google page-rank algorithm we saw in chapter 4.

在诸如此类的复杂问题中,以及在物理和数学中,当方程式太难精确求解时,或者当您想要模拟某些事物(例如两个黑洞的合并或未来的气候情景)时,通常需要使用数值方法。在最简单的层面上,这些方法涉及猜测一个解决方案,然后使用算法调整该解决方案,从而得到越来越接近的近似值。牛顿是这种方法的先驱之一,但今天有一个专门研究复杂计算数学的领域,包括数值线性代数 (NLA)。张量或基于张量思想的算法在许多这些计算方法中以各种方式用于解决令人眼花缭乱的问题。4

In complex problems such as these—as well as in physics and maths when equations are too difficult to solve exactly, or where you want to simulate something such as the merging of two black holes, or a future climate scenario—it is often necessary to use numerical methods. At the simplest level, these methods involve guessing a solution, and then using an algorithm that tweaks this solution, giving a closer and closer approximation. Newton was one of the pioneers of this approach, but today there’s a whole field devoted to sophisticated computational maths, including numerical linear algebra (NLA). Tensors, or algorithms based on tensorial ideas, are used in many of these computational methods in a variety of ways to solve a dizzying array of problems.4

在许多现代应用中,张量思想不仅在传统的里奇微积分意义上得到使用,而且经常以巧妙的新方式得到改编。例如,在第 11 章中,我们介绍了数据科学中的“不规则张量”思想。这种改编的另一个例子是狄拉克的惊人发现,即他的场方程只有在系数是矩阵而不是像度量张量的分量g μν这样的普通函数时才具有物理意义。我们在这个故事中一次又一次地看到了这种思想的演变:数学家从数字到向量的方式,古老的表格演变成矩阵的方式,以及然后向量和矩阵演变成张量;代数发展成函数微积分,然后发展成向量和张量的微积分;数学家从代数变换方程发展到张量算子。

In many of these modern applications, tensor ideas are not only used in the traditional sense of Ricci’s calculus but are often adapted in clever new ways. For instance, in chapter 11 we saw the idea of “ragged tensors” in data science. A different example of this kind of adaptation is Dirac’s remarkable realisation that his field equations could only make physical sense if the coefficients were matrices, rather than ordinary functions like the metric tensor’s components gμν. We’ve seen this kind of evolution of ideas over and over in this story: in the way mathematicians went from numbers to vectors, and the way ancient tables morphed into matrices, and then vectors and matrices morphed into tensors; the way algebra developed into the calculus of functions, and then into the calculus of vectors and tensors; and the way mathematicians went from algebraic transformation equations to tensor operators.

再举一个张量思想改编的例子,张量网络 (TN) 使用张量的现代思想作为多线性映射——我在第 11 章中提到过这个想法,当时我谈到张量的现代定义是操作向量和一元形式以得到标量的东西。TN 用于映射和标记像素阵列以进行图像分类,或者对纠缠属性进行分类,仅举两个例子——我们已经看到张量索引对于标记不同特征有多么有用。为了处理大量数据,TN 还使用矩阵方法的张量改编,例如我在第 4 章中提到的分解。

To take just one more example of the adaptation of tensor ideas, tensor networks (TN) use the modern idea of tensors as multilinear maps—an idea I implied in chapter 11, when I spoke of the modern definition of a tensor as something that operates on vectors and one-forms to give a scalar. TNs are used to map and label arrays of pixels for image classification, say, or to classify entanglement properties, to take just two examples—and we’ve seen how useful tensor indices are for labeling different features. To handle huge amounts of data, TNs also use tensor adaptations of matrix methods such as the decomposition I mentioned in chapter 4.

奇妙的感觉

A SENSE OF WONDER

事实上,我们在这个故事中提到的所有数学工具在今天都是不可或缺的:代数、微积分、虚数、向量、四元数、矩阵和张量。它们所实现的技术有助于让我们的生活更加舒适和有趣,我不想没有它。当然,和我们今天的许多人一样,我确实希望它得到更好的监管,因为我们现在知道,当技术被误用或使用时不照顾人类和地球时,它就有其阴暗的一面。

In fact, all the mathematical tools we’ve followed in this story are indispensable today: algebra, calculus, imaginary numbers, vectors, quaternions, matrices, and tensors. The technology they’ve enabled helps make our lives more comfortable and interesting, and I wouldn’t want to be without it. Of course, like many of us these days I do want it to be better regulated, for we now know that technology has its dark side when it’s misapplied or used without taking care of people and the planet.

但是,如果我们想要过上有意义的生活,我们需要的不仅仅是高科技的舒适感。我们还需要一种好奇心。我们的祖先敬畏地凝视着夜空十万年。他们在美丽的风景中寻求慰藉,他们学会了如何与地球共存,并从地球中汲取营养。他们也很好奇,试图了解大自然的秘密——以及如何利用这些秘密知识让生活变得更轻松。所有这些都让我们成为人类。但正如我们现在所清楚地知道的那样,在这一过程中的某个时刻,正是这些科学知识——以及对“舒适”的不断增长的渴望——在发达国家,这还意味着拥有大量的“东西”和赚取巨额利润——破坏了大自然微妙的平衡。我们砍伐了太多的森林,毒害了太多的河流,烧毁了太多的化石燃料,让我们失去了地球上所有美丽和奇妙的东西。我们正在失去人类不可或缺的一部分。

We need more than high-tech comfort, though, if we want a meaningful life. We need a sense of wonder, too. Our ancestors gazed in awe at the night sky for a hundred thousand years. They sought solace in a beautiful landscape, and they learned how to live with and from the Earth. They were curious, too, trying to understand nature’s secrets—and how to use this secret knowledge to make life easier. All this is what makes us human. But as we now know all too starkly, somewhere along the way that very scientific knowledge—and that ever-increasing desire for “comfort,” which in the developed world now also means owning lots of “stuff ” and making huge profits—helped disrupt the delicate balance of nature. We’ve cut down too many forests, poisoned too many rivers, and burned too many fossil fuels, so that we are losing all that was beautiful and wondrous about our planet. We are losing an essential part of our humanity.

然而,大自然既美丽又可怕,科学帮助我们理解并减轻了部分恐惧。例如,科学解释了曾经被视为神灵不祥预兆的日食,它提供天气预报帮助我们预防洪水和火灾,提供电力保护我们免受黑暗和寒冷。虽然生产电力的后果使大自然更加可怕,但科学和技术已成为应对灾难性气候变化的关键武器,而令人振奋的绿色未来就在前方。当然,这个未来不能只关乎技术。我们也需要与大自然的古老联系。尽管如此,这些新技术充满了人类智慧的奇妙例子,也将我们与过去联系在一起。

Yet nature can be terrible as well as beautiful, and it was science that helped us to understand and mitigate some of that terror. It explained eclipses that once seemed malign portents from the gods, for example, and it makes weather forecasts to help us prepare for floods and fires, and electricity to protect us from the dark and cold. And while the effects of producing that electricity have now made nature even more terrifying, science and technology have become key weapons against cataclysmic climate change, and there is an exciting, greener future ahead. This future cannot be all about technology, of course. We need our ancient link with nature, too. Still, these new technologies are full of marvelous examples of human ingenuity that also connect us with our past.

确实,数学和数学科学这一宏伟的智力创造中充满了奇迹。我写这本书就是为了与你们分享其中的一些想法,并展示这个故事中的数学概念是如何在人类想象力的五千年多元文化之旅中演变而来的。这样的旅程将我们与先辈联系在一起,就像惊叹于星空一样。

Indeed, there is wonder aplenty in the magnificent intellectual creation that is mathematics and mathematical science. I’ve written this book to share some of these ideas with you—and to show how the mathematical concepts in this story have evolved over their extraordinary five-thousand-year multicultural journey through the human imagination. Such an odyssey connects us with our forerunners, just as marveling at a starry sky does.

当爱因斯坦终于找到他的场方程时,他告诉一位朋友,他认为这些方程“美得无与伦比”。5在这个故事中,我们看到了一些美丽的方程的诞生。爱因斯坦的方程是里奇张量分析的最高成就,我们还看到了麦克斯韦方程如何演变成它们美丽的矢量和张量形式。我们看到,汉密尔顿在布鲁姆桥上刻下的四元数规则非常经济,欧拉方程本身就优雅而简单,牛顿定律非常简洁有力,等等。我认为能够欣赏这种智力之美很重要,就像欣赏音乐和艺术之美一样重要。

When Einstein finally found his field equations, he told a friend he thought they were “beautiful beyond comparison.”5 In this story we’ve seen some beautiful equations come into being. Einstein’s were the crowning achievement of Ricci’s tensor analysis, and we also saw how Maxwell’s equations evolved into their beautiful vector and tensor forms. We saw that Hamilton’s quaternion rules scratched on Broome Bridge are delightfully economical, Euler’s equation is elegant simplicity itself, Newton’s laws are amazingly concise and potent, and much more. I think it is important to be able to appreciate this kind of intellectual beauty, just as it is important to appreciate beauty in music and the arts.

我们也遇到了一些迷人而又敬业的人。他们中的一些人主要是出于解决实际问题的愿望——尽管我们最近的技术发展过于激进,但我想在这个故事中提出的一件事是数学思想是如何演变的兼顾这些实际需要。另一方面,许多先驱者只是出于了解和理解的渴望,或出于追随有趣的模式或证据的兴奋。这个故事表明我们需要这两种思想家。多亏了他们,我们人类不仅有可能过上安全、舒适和有趣的生活,而且我们还可以理解如此多关于我们广阔而令人费解的宇宙的事情,这真是太棒了。这就是数学的力量!

We’ve met some fascinating and dedicated people, too. Some of them were motivated primarily by the desire to solve practical problems—and, despite our recent technological overreach, one of the things I’ve wanted to bring out in this story is the way mathematical ideas have evolved hand in hand with these practical needs. On the other hand, many of these pioneers were driven simply by the desire to know and understand, or by the thrill of following through an intriguing pattern or proof. This story has shown that we need both kinds of thinker. It is truly awesome that, thanks to all of them, we humans not only have the possibility of safe, comfortable, and interesting lives; we can also comprehend so much about our vast and mind-bending universe. Such is the power of mathematics!

时间线

TIMELINE

约公元前 3000 年,美索不达米亚和埃及:制作了已知的第一份楔形文字和象形文字数据表。

ca. 3000 BCE, Mesopotamia and Egypt: The first known cuneiform and hieroglyphic tables of data are made.

约公元前 2000 年,美索不达米亚:楔形文字乘法表和文献显示了农田面积的计算;使用几何方法通过完成平方来求解二次方程。

ca. 2000 BCE, Mesopotamia: Cuneiform multiplication tables and documents show calculations for the area of agricultural fields; geometric methods are used to solve quadratic equations by completing the square.

约公元前 1700 年,美索不达米亚:Plimpton 322 证明毕达哥拉斯定理在毕达哥拉斯之前就已经为人所知。

ca. 1700 BCE, Mesopotamia: Plimpton 322 demonstrates Pythagoras’s theorem is known before Pythagoras.

约公元前 1650 年,埃及:阿赫梅斯纸莎草纸中包含了求圆的周长和面积的近似方法。

ca. 1650 BCE, Egypt: Ahmes papyrus includes approximate methods for finding the circumference and area of a circle.

约公元前 300 年,埃及/希腊:《欧几里得几何原本》给出了平面(“欧几里得”)空间的几何规则。

ca. 300 BCE, Egypt/Greece: Euclid’s Elements gives the rules of geometry governing flat (“Euclidean”) space.

约公元前 250 年,中国:九章算术》包含了今天称为高斯消元法的算法。

ca. 250 BCE, China: Jiuzhang Suanshu (Nine Chapters on the Mathematical Art) includes the algorithmic method known today as Gaussian elimination.

约公元前 200 年,西西里岛/希腊:阿基米德接近微积分的思想。

ca. 200 BCE, Sicily/Greece: Archimedes comes close to the idea of calculus.

约公元 150 年,埃及/希腊:托勒密借鉴了前辈(尤其是埃拉托色尼和阿里斯塔克斯)的作品,撰写了《天文学大成》《地理学》,这两本书共同构成了一部里程碑式的著作数学天文学、三角学和坐标(天体和地面的纬度和经度)的编纂。三角学和坐标是创建矢量的基础。

ca. 150 CE, Egypt/Greece: Ptolemy, drawing on forerunners whose works are no longer extant—notably Eratosthenes and Aristarchus— writes his Almagest and Geography, which together make a landmark compilation of mathematical astronomy, trigonometry, and the use of coordinates (celestial and terrestrial latitude and longitude). Trigonometry and coordinates will be fundamental in the creation of vectors.

约公元 400 年,埃及/希腊:希帕提娅撰写了《天文学大成》和其他著作的博学评论;公元 415 年,希帕提娅被狂热分子残忍杀害。公元七世纪,印度:数学蓬勃发展;婆罗摩笈多同时考虑正数和负数。

ca. 400 CE, Egypt/Greece: Hypatia writes learned commentaries on Almagest and other works; 415 CE, Hypatia is brutally murdered by zealots. Seventh century CE, India: Mathematics flourishes; Brahmagupta considers both positive and negative numbers.

公元 9 世纪:阿拉伯数学蓬勃发展,包括巴格达哈里发马蒙智慧之家的穆罕默德·伊本·穆萨·花拉子密;约公元 830 年,花拉子密撰写了教科书《代数》(后来“代数”一词由此而来)。但代数仍然使用文字进行,因为代数符号直到 17 世纪才出现。

Ninth century CE: Arabic mathematics flourishes, including Mohammed ibn-Mūsā al-Khwārizmī at Caliph al-Ma’mūn’s House of Wisdom in Baghdad; ca. 830 CE, al-Khwārizmī writes the textbook Al-Jabr wa’l muqābalah (from which the name “algebra” is later derived). But algebra is still done using words, for algebraic symbolism will not exist until the seventeenth century.

约 1200 年,中东(现代伊朗):沙拉夫丁·塔西 (Sharaf al-Dīn al-Ṭūsī) 在解决三次方程方面做出了开创性的工作。

ca. 1200, Middle East (modern Iran): Sharaf al-Dīn al-Ṭūsī does pioneering work on solving cubic equations.

1540 年,意大利:吉罗拉莫·卡尔达诺的大艺术》包含了解决三次方程的首个算法。他对其中一些解法中出现的“虚数”感到困惑。

1540, Italy: Girolamo Cardano’s Ars Magna (The Great Art) contains the first algorithms for solving cubic equations. He is flummoxed by the “imaginary” numbers showing up in some of these solutions.

1572 年,意大利:拉斐尔·邦贝利(Rafael Bombelli)的《代数学》开创了复数数学的先河;这些数字在四元数和向量的发现中发挥了关键作用。

1572, Italy: Rafael Bombelli’s Algebra pioneers the math of complex numbers; these numbers will play a key role in the discovery of quaternions and vectors.

1619年,英国:托马斯·哈里奥特在碰撞力学方面的工作预示了矢量的概念,并正确地使用了平行四边形规则。

1619, England: Thomas Harriot’s work on the mechanics of collisions prefigures the idea of vectors and uses the parallelogram rule correctly.

1631 年,英国:托马斯·哈里奥特(Thomas Harriot)的《分析艺术实践》 (Artis Analyticae Praxis)在他死后出版——这是第一本方程式完全由符号组成且基本上使用现代符号主义的代数教科书。

1631, England: Thomas Harriot’s Artis Analyticae Praxis (Practice of the Analytical Art) is posthumously published—the first algebra textbook where the equations are fully symbolic and use essentially modern symbolism.

1637 年,法国:勒内·笛卡尔的《方法论》引入了代数符号xy,并开创了“笛卡尔”坐标。

1637, France: René Descartes’s Discourse on Method introduces the algebraic symbols x and y and pioneers what would become “Cartesian” coordinates.

17 世纪 70 年代,英国/德国:艾萨克·牛顿和戈特弗里德·莱布尼茨分别独立创建了通用的算法微分和积分学。

1670s, England/Germany: Isaac Newton and Gottfried Leibniz independently create general, algorithmic differential and integral calculus.

1687 年:英国:牛顿出版他的自然哲学原理Mathematica自然哲学的数学原理);它包括他的一般微积分算法和他的引力理论,并向物理学引入了力和速度是矢量的思想。然而,他的大部分证明都是几何学的,因为他(正确地)认为微积分还不够严格。(它需要 19 世纪将出现的极限和连续性的定义。)

1687: England: Newton publishes his Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy); it includes his general calculus algorithms and his theory of gravity and introduces to physics the idea that forces and velocities are vectorial. Most of his proofs, however, are done geometrically because he (rightly) feels calculus is not yet rigorous enough. (It needs the definitions of limits and continuity that will come in the nineteenth century.)

18 世纪初,瑞士:约翰·伯努利(又名让·伯努利)在莱布尼茨与牛顿关于微积分优先权的争论中为莱布尼茨辩护;后来他开始将牛顿的证明转化为代数(但莱布尼茨式的)微积分。

Early 1700s, Switzerland: Johann (aka Jean) Bernoulli defends Leibniz in the calculus priority dispute with Newton; later he begins to translate Newton’s proofs into algebraic (but Leibnizian) calculus.

1759 年,法国:埃米莉·杜·夏特莱 (Émilie du Châtelet) 的《自然哲学的数学原理》法语译本出版——这是英国以外的第一个译本,至今仍是权威的法语版本。该译本包含一个技术附录,她在其中帮助开创了用现代(莱布尼茨)微积分编写牛顿几何证明的过程。

1759, France: Émilie du Châtelet’s French translation of Principia is published—the first translation outside Britain and still the definitive French version today. It includes a technical appendix, in which she helps pioneer the process of writing Newton’s geometrical proofs in terms of modern (Leibnizian) calculus.

1785年,法国:查尔斯·库仑证明静电荷的作用力遵循平方反比定律,就像牛顿万有引力定律一样。

1785, France: Charles Coulomb shows that the force from a static electric charge obeys an inverse square law like Newton’s law of gravity.

18 世纪 80 年代,法国:约瑟夫-路易·拉格朗日根据引力“势”重新阐述了牛顿引力定律。爱因斯坦将在自己的引力理论(广义相对论)中使用这一思想。

1780s, France: Joseph-Louis Lagrange reformulates Newton’s law of gravity in terms of a gravitational “potential.” Einstein will use this idea in his own theory of gravity (general relativity).

1788 年,法国:拉格朗日出版了《分析力学》 (Mécanique Analytique),这本书对更新牛顿力学并将其表达为现代微积分形式大有裨益。半个世纪后,赫尔曼·格拉斯曼将受到这本书的启发,开始他独特的旅程,共同发现矢量分析。

1788, France: Lagrange publishes Mécanique Analytique (Analytical Mechanics), which goes a long way toward updating Newtonian mechanics and expressing it in modern calculus form. Half a century later, Hermann Grassmann will be inspired by this book, when he sets out on his idiosyncratic journey to codiscovering vector analysis.

1799 年,法国:皮埃尔-西蒙·拉普拉斯出版了他的巨著《天体力学论》第一卷。这也启发了格拉斯曼,并对威廉·罗恩·汉密尔顿同时期的矢量分析发展产生了影响。它还将启发自学成才的玛丽·萨默维尔,她于 1831 年出版了拉普拉斯前两卷著作《天体力学》的详尽英文版。

1799, France: Pierre-Simon Laplace publishes the first volume of his monumental Traité de Mécanique Céleste (Treatise on Celestial Mechanics). This also inspires Grassmann and is influential on William Rowan Hamilton’s contemporaneous development of vector analysis. It will also inspire the self-taught Mary Somerville, whose explicated, expanded English version of Laplace’s first two volumes, Mechanism of the Heavens, will be published in 1831.

1790 年代,法国:拉格朗日和拉普拉斯是引入公制的度量衡委员会成员。

1790s, France: Lagrange and Laplace are members of the Committee on Weights and Measures that introduces the metric system.

1800 年,意大利:亚历山德罗·伏特 (Alessandro Volta) 发明了第一个电池,首次产生了电流。

1800, Italy: Alessandro Volta invents the first electric battery, enabling electric currents to be produced for the very first time.

1801年,英国:托马斯·杨的双缝实验揭示了光的波动性。

1801, England: Thomas Young’s double-slit experiment shows the wave nature of light.

19 世纪初,欧洲:阿尔冈(法国)、高斯(德国)、韦塞尔(挪威)和沃伦(英国)分别在平面上表示复数。乘以i很快就被解释为围绕该平面旋转 90°。

Early 1800s, Europe: Argand (France), Gauss (Germany), Wessel (Norway), and Warren (England) independently represent complex numbers on a plane. Multiplication by i is soon interpreted as a 90° rotation around this plane.

19 世纪初,法国:索菲·热尔曼与卡尔·高斯通信;1816 年,她因一篇开创性地研究振动表面数学的论文获得了法国科学院颁发的著名数学奖。

Early 1800s, France: Sophie Germain corresponds with Carl Gauss; in 1816 she wins a prestigious mathematical prize from the French Academy of Sciences, for a paper pioneering the mathematics of vibrating surfaces.

1820 年,丹麦:汉斯·奥斯特 (Hans Øersted) 发现变化的电流会对磁铁产生影响。

1820, Denmark: Hans Øersted discovers that a changing electric current affects a magnet.

1821 年,法国:安德烈-玛丽·安培利用 Øersted 的发现发明了第一个电报系统。他后来成为电磁理论和实验的重要贡献者。

1821, France: André-Marie Ampère applies Øersted’s discovery by inventing the first telegraphic system. He goes on to be an important contributor to electromagnetic theory and experiment.

1821 年,英国:迈克尔·法拉第应用奥斯特的发现,发明了第一台电动机原型。

1821, England: Michael Faraday applies Øersted’s discovery by inventing the first prototype of an electric motor.

1822 年,法国:约瑟夫·傅立叶推导出热方程;他在 1827 年的一篇论文中运用了该方程,从此开创了气候科学。

1822, France: Joseph Fourier derives the heat equation; he applies it in an 1827 paper that initiates climate science.

1828年,德国:卡尔·弗里德里希·高斯发展非欧几里得几何。(匈牙利的亚诺什·波尔艾和俄国的尼古拉·罗巴切夫斯基也在做这方面的工作。)高斯定义了曲面的本征几何,定义了曲面上的距离测度(线元或度规),以及曲面的本征曲率。

1828, Germany: Carl Friedrich Gauss develops non-Euclidean geometry. ( Janos Bolyai in Hungary and Nicolai Lobachevsky in Russia are also working on this.) Gauss defines a curved surface’s intrinsic geometry, defining the distance measure (line element or metric) on a curved surface, and the intrinsic curvature of the surface.

1831年,英国:迈克尔·法拉第发现运动的磁铁可以产生电力,并发明了发电机的原型。法拉第和奥斯特的发现就是电磁学的发现。随后,法拉第提出了电场和磁场的概念。

1831, England: Michael Faraday discovers that a moving magnet can generate an electric force; as a result, he invents the prototype of the electric generator. Faraday’s and Øersted’s discoveries amount to the discovery of electromagnetism. Subsequently, Faraday conceives the idea of electric and magnetic fields.

1831 年,苏格兰/英格兰:玛丽·萨默维尔的天文学教科书《天体机制》受到顶尖科学家的热烈欢迎。她使用了莱布尼茨微积分。

1831, Scotland/England: Mary Somerville’s astronomy textbook Mechanism of the Heavens is rapturously received by leading scientific men. She uses Leibnizian calculus.

1832 年,爱尔兰:威廉·罗恩·汉密尔顿利用微积分预测圆锥折射,这是对以前未知的物理现象的最早的数学预测之一。

1832, Ireland: William Rowan Hamilton uses calculus to predict conical refraction, one of the very first mathematical predictions of a previously unknown physical phenomenon.

1833 年、1840 年,英国:威廉·惠威尔 (William Whewell) 创造了“科学家”和“物理学家”这两个术语。

1833, 1840, England: William Whewell coins the terms “scientist” and “physicist.”

1837 年,爱尔兰:汉密尔顿被任命为爱尔兰皇家学院院长;他希望将文学和科学纳入其中,并向著名作家玛丽亚·埃奇沃思寻求建议。他采纳了大部分建议,但没有听从她让女性参加会议的建议。

1837, Ireland: Hamilton is made president of the Royal Irish Academy; he wants to include literature as well as science and asks the venerable writer Maria Edgeworth for advice. He takes most of it but doesn’t heed her suggestion to make meetings available to women.

1843 年,爱尔兰:汉密尔顿发明了四元数和向量;1843 年 10 月 16 日,他在布鲁姆桥上划出了它们的基本规则。一个月后,他在爱尔兰皇家学院宣读的一篇论文中正式宣布了他的发现。

1843, Ireland: Hamilton invents quaternions and vectors; October 16, 1843, he scratches their basic rules on Broome Bridge. A month later he formally announces his discovery in a paper read to the Royal Irish Academy.

1843 年,英国:威廉·华兹华斯被任命为桂冠诗人。他与热爱诗歌并创作诗歌的汉密尔顿是好朋友。

1843, England: William Wordsworth is made poet laureate. He is friendly with Hamilton, who loves and composes poetry.

1844 年,德国:赫尔曼·格拉斯曼(Hermann Grassmann)出版了他的关于我们现在所说的向量和向量空间的书《Ausdehnungslehre》

1844, Germany: Hermann Grassmann publishes his book on what we now call vectors and vector spaces, Ausdehnungslehre.

19 世纪 40 年代初,英国:阿瑟·凯莱 (Arthur Cayley) 与乔治·布尔 (George Boole) 就不变量进行通信;凯莱哀叹他们之间缺乏运输手段(第一条铁路仍在铺设中)。

Early 1840s, England: Arthur Cayley corresponds with George Boole on invariants; Cayley laments the lack of transport between them (the first railways are still being laid).

1845 年,美国:作家亨利·戴维·梭罗在瓦尔登湖畔的树林中寻求更简单的生活。他的著作《瓦尔登湖》有助于开创环保运动,并展示个人如何使用更少的资源。

1845, United States: Writer Henry David Thoreau seeks a simpler life in the woods by Walden Pond. His book Walden will help pioneer the environmental movement and show how individuals can use fewer resources.

19 世纪 50 年代,美国/爱尔兰:尤妮丝·牛顿·富特和约翰·廷德尔分别独立开创了二氧化碳加热(温室)效应的科学。

1850s, United States/Ireland: Eunice Newton Foote and John Tyndall independently pioneer the science of carbon dioxide’s heating (greenhouse) effect.

1855 年,苏格兰/英格兰:詹姆斯·克拉克·麦克斯韦发表了第一篇关于法拉第电场和磁场思想的论文。他和威廉·汤姆森(麦克斯韦的论文就是取自他的论文)是唯一做出重大尝试将法拉第思想转化为数学(矢量)语言的人。

1855, Scotland/England: James Clerk Maxwell publishes his first paper on Faraday’s idea of electric and magnetic fields. He and William Thomson (on whose paper Maxwell drew) are the only ones to make significant attempts at putting Faraday’s idea into mathematical (vectorial) language.

1857 年,苏格兰/爱尔兰:彼得·格思里·泰特(麦克斯韦的童年好友)开始应用汉密尔顿的矢量微积分算子 ∇ 和(与在助手的帮助下,他将其命名为“nabla”。他和汉密尔顿很快就开始互相通信。

1857, Scotland/Ireland: Peter Guthrie Tait (Maxwell’s childhood friend) begins to apply Hamilton’s vector calculus operator, ∇, and (with the help of his assistant) names it “nabla.” He and Hamilton soon begin corresponding with each other.

1858 年,英国:阿瑟·凯莱 (Arthur Cayley) 形式化了矩阵代数;他受到汉密尔顿 (Hamilton) 发现非交换四元数代数的启发。

1858, England: Arthur Cayley formalises the algebra of matrices; he has been inspired by Hamilton’s discovery of noncommutative quaternion algebra.

19 世纪 50 年代末,苏格兰/英格兰:威廉·汤姆森(未来的开尔文勋爵,麦克斯韦和泰特的朋友)帮助在大西洋下铺设了第一条海底电报电缆——他是北大西洋电报公司的董事和科学顾问。

Late 1850s, Scotland/England: William Thomson (the future Lord Kelvin, and a friend of Maxwell and Tait) helps lay the first submarine telegraphic cable under the Atlantic Ocean—he is a director and scientific advisor of the North Atlantic Telegraph Company.

1854 年,1861 年,德国:伯恩哈德·黎曼将他以前的教授高斯的曲率分析应用到任意维度的空间,超出了我们习惯的三维空间。他发现识别曲率的条件可以用后来被称为黎曼张量的张量来表达。(如果表面是平的,黎曼张量为零,反之亦然。)

1854, 1861, Germany: Bernhard Riemann applies his former professor Gauss’s analysis of curvature to spaces of arbitrary dimension, beyond the usual three we’re used to. He finds the condition for identifying curvature is expressed in what will become known as the Riemann tensor. (If the surface is flat, the Riemann tensor is zero, and vice versa.)

1865 年 1 月,英国:麦克斯韦于 1864 年底首次提出的电磁场理论发表。该理论是矢量的,但方程是分量形式,尚未形成完整的全矢量微积分形式。麦克斯韦从他的方程中推断出光是一种电磁波,并暗示可能存在其他电磁辐射。

January 1865, England: Maxwell’s theory of the electromagnetic field, first presented at the end of 1864, is published. It is vectorial, but the equations are in component form, not yet in full, whole-vector calculus form. Maxwell deduces from his equations that light is an electromagnetic wave and suggests the possible existence of other electromagnetic radiation.

1867 年,苏格兰/英格兰:汤姆森和泰特出版了他们成功的物理教科书《自然哲学论文》(俗称TT′);其中没有四元数,因为汤姆森认为四元数和整个向量是无用的,因为它们是计算所需的分量。同年,泰特出版了他的《四元数初等论文》,进一步发展了汉密尔顿的向量微积分。

1867, Scotland/England: Thomson and Tait publish their successful physics textbook A Treatise on Natural Philosophy (known colloquially as T and T′); it has no quaternions in it, because Thomson thinks quaternions and whole vectors are useless since it is the components one needs to calculate with. In the same year, Tait publishes his Elementary Treatise on Quaternions, which further develops Hamilton’s vector calculus.

1870 年,苏格兰:麦克斯韦开创了矢量微积分术语“散度”、“梯度”和“旋度”。

1870, Scotland: Maxwell pioneers the vector calculus terms “divergence,” “grad,” and “curl.”

1873 年,英国:麦克斯韦的《电磁论》出版;第 2 卷包括他的电磁方程的全矢量形式(四元数符号)。它还定义了后来被称为应力张量的量纲;柯西和汤姆逊做了类似的事情,但麦克斯韦给了它一个双索引符号,后来成为标准。

1873, England: Maxwell’s Treatise on Electricity and Magnetism is published; volume 2 includes the whole-vector form of his equations of electromagnetism (in quaternionic notation). It also defines what would become known as the stress tensor; Cauchy and Thomson had done something similar, but Maxwell gave it a two-index symbol, which later would become standard.

19 世纪 70 年代,英国:威廉·金登·克利福德 (William Kingdon Clifford) 在综合了汉密尔顿和格拉斯曼的向量创造的基础上,创建了一种“几何代数”。他与泰特和麦克斯韦以及著名小说家乔治·艾略特关系密切。(她的代表作《米德尔马契》刚刚出版,出版时间是 1871-72 年。)

1870s, England: William Kingdon Clifford creates a “geometric algebra” based on a synthesis of Hamilton’s and Grassmann’s creation of vectors. He is friendly with Tait and Maxwell and with the famous novelist George Eliot. (Her masterwork Middlemarch has just been published, in 1871–72.)

1880 年代,英国/美国:奥利弗·赫维赛德和约西亚·吉布斯各自创立了现代矢量分析。他们的灵感首先来自麦克斯韦在《电磁论》中使用矢量描述电磁学,然后是汉密尔顿和泰特关于四元数的著作(他们从麦克斯韦给出的参考文献中找到)。

1880s, England/United States: Oliver Heaviside and Josiah Gibbs independently create modern vector analysis. Their inspiration is, first, Maxwell’s use of vectors to describe electromagnetism in his Treatise, and then Hamilton and Tait’s works on quaternions (which they found from the references Maxwell gave).

19 世纪 80 年代,意大利:格雷戈里奥·里奇 (Gregorio Ricci) 开发了他的“绝对微分学”,即如今的张量微积分。吉布斯 (Gibbs) 扩展了格拉斯曼的工作,也提出了张量积的概念。与整个向量一样,主流数学家看到了张量的优雅之处,但没有看到其实用性。

1880s, Italy: Gregorio Ricci develops his “absolute differential calculus,” which is known today as tensor calculus. Gibbs, extending Grassmann’s work, also has the idea of tensor products. As with whole vectors, mainstream mathematicians see the elegance but not the practical point of tensors.

1888 年,德国/英国:海因里希·赫兹和奥利弗·洛奇宣布首次有意创造无线电波,惊人地证实了麦克斯韦对其存在的数学预测。

1888, Germany/England: Heinrich Hertz and Oliver Lodge announce the first-ever deliberate creation of radio waves, spectacularly confirming Maxwell’s mathematical prediction of their existence.

1888 年,意大利:朱塞佩·皮亚诺 (Giuseppe Peano) 设计了向量空间的现代公理定义。

1888, Italy: Giuseppe Peano devises the modern axiomatic definition of a vector space.

19 世纪 80 年代末至 90 年代初,英国、美国、澳大利亚:矢量“战争”:Heaviside 和 Gibbs 主张矢量分析,Tait、Alexander McAulay 等人则主张四元数;Cayley 和 Thomson 则与他们所有人争辩,他们更倾向于分量而不是整个矢量和四元数。矢量分析(实数、整个矢量的数学)获胜(直到 20 世纪末四元数再次出现)。

Late 1880s–early 1890s, United Kingdom, United States, Australia: The Vector “Wars”: Heaviside and Gibbs for vector analysis vs. Tait, Alexander McAulay, and others for quaternions; Cayley and Thomson vs. all of them, favouring components over whole vectors and quaternions. Vector analysis—the math of real, whole vectors—wins (until quaternions show up again in the late twentieth century).

19 世纪 90 年代末,瑞士:同班同学阿尔伯特·爱因斯坦和米列娃·马里奇成为恋人。马里奇和同学马塞尔·格罗斯曼是第一批认可和支持爱因斯坦天才的人。格罗斯曼后来成为一名成功的数学教授,但班上唯一的女生马里奇没有资格毕业(性别歧视?);她再次尝试,但最终失去了成为一名科学家的梦想。

Late 1890s, Switzerland: Albert Einstein and Mileva Marić, who are in the same physics class, become lovers. Marić and fellow student Marcel Grossmann are the first to recognise and support Einstein’s genius. Grossmann goes on to a successful career as a mathematics professor, but Marić, the only girl in their class, does not qualify to graduate (sexism?); she tries again, but ultimately loses her hard-fought dream of becoming a scientist.

1895 年,德国:格蕾丝·奇泽姆 (Grace Chisholm) 获得了剑桥大学的非正式学位,随后在哥廷根与费利克斯·克莱因 (Felix Klein) 一起攻读数学博士学位,她是德国第一位获得正式博士学位的女性,也是世界上首批获得正式博士学位的女性之一。女性直到 1920 年才能够获得牛津大学的正式学位,直到 1948 年才能够获得剑桥大学的正式学位。

1895, Germany: Grace Chisholm, who received an unofficial degree from Cambridge and then worked on her doctorate in mathematics at Göttingen with Felix Klein, is the first woman to receive an official doctorate in Germany, and one of the first anywhere in the world. Women will not be able to take official degrees from Oxford until 1920 or from Cambridge until 1948.

1900 年,意大利/德国:应里奇的前导师、 《数学年鉴》主编菲利克斯·克莱因的要求,格雷戈里奥·里奇和他的前学生图利奥·列维-奇维塔撰写了一篇关于里奇“绝对微分学”(今天称为张量微积分)的概述,里奇和列维-奇维塔的论文就是在该杂志上发表的。十二年后,它将启发爱因斯坦和格罗斯曼寻找广义相对论的数学基础。

1900, Italy/Germany: Gregorio Ricci and his former student Tullio Levi-Civita write an overview of Ricci’s “absolute differential calculus”— today called tensor calculus—at the request of Felix Klein, Ricci’s former mentor and an editor of Mathematische Annalen, where the Ricci–Levi-Civita paper is published. Twelve years later, it will inspire Einstein and Grossmann in their search for the mathematical foundations of the general theory of relativity.

1905 年,瑞士:爱因斯坦在担任专利官员期间发表了五篇开创性的论文,其中包括《论动体的电动力学》,今天人们更熟悉的是,这篇论文创立了狭义相对论。亨德里克·洛伦兹 (1904) 和亨利·庞加莱 (1905) 也有类似的理论,但它们是基于以太的,并不是完全相对论的。洛伦兹使用全矢量符号,而庞加莱和爱因斯坦使用分量符号。

1905, Switzerland: Einstein, working as a patent officer, publishes five groundbreaking papers, including On the Electrodynamics of Moving Bodies, better known today as the paper that created the special theory of relativity. Hendrik Lorentz (1904) and Henri Poincaré (1905) also have similar theories, but they are ether-based and are not fully relativistic. Lorentz uses whole-vector notation, but Poincaré and Einstein use components.

1907年,德国:爱因斯坦的前数学讲师赫尔曼·闵可夫斯基利用狭义相对论创建了时空概念。

1907, Germany: Hermann Minkowski, Einstein’s former maths lecturer, uses the special theory of relativity to create the concept of space-time.

1908-10 年,德国:闵可夫斯基开始以张量形式写出麦克斯韦方程,闵可夫斯基突然去世后,他的好友阿诺德·索末菲接手了这项工作。他们创造了能量动量张量的概念,这将成为广义相对论的关键。

1908–10, Germany: Minkowski begins to write Maxwell’s equations in tensor form, a task then taken up after Minkowski’s sudden death by his friend Arnold Sommerfeld. They create the concept of the energymomentum tensor, which will be a key to the general theory of relativity.

1912 年,瑞士:爱因斯坦和他的老朋友格罗斯曼现在是他们母校瑞士理工学院(简称 ETH)的同事。他们发现了里奇的张量微积分,并合作研究了广义相对论的数学基础。

1912, Switzerland: Einstein and his old friend Grossmann are now colleagues at their old school, the Swiss Polytechnic or ETH. They discover Ricci’s tensor calculus and collaborate on the mathematical foundations of general relativity.

1914-18 年,第一次世界大战:除了前线的恐怖之外,食物也匮乏,科学交流也中断(包括爱因斯坦和列维-奇维塔之间关于张量理论的交流)。

1914–18, World War I: In addition to the horrors on the front, food is scarce, and scientific communication is interrupted (including that between Einstein and Levi-Civita on tensor theory).

1914 年,德国:爱因斯坦在柏林担任教授。爱因斯坦和马里奇的婚姻破裂。爱因斯坦一直非常努力地发展广义相对论。他还在追求他的表妹艾尔莎·爱因斯坦。

1914, Germany: Einstein takes up a professorship in Berlin. The Einstein-Marić marriage breaks down. Einstein has been working incredibly hard developing the general theory of relativity. He is also courting his cousin Elsa Einstein.

1915 年,德国:大卫·希尔伯特和爱因斯坦合作(并最终竞争)完成广义相对论的最后阶段。该理论于 11 月完成。经过一些小摩擦后,两人仍然是友好的同事。爱因斯坦创造了时空曲率的概念来表示引力的影响,并在格罗斯曼的帮助下,他用张量实现了这一概念。今天,广义相对论仍然是世界智力奇迹之一。

1915, Germany: David Hilbert and Einstein collaborate (and ultimately compete) on the final steps toward the general theory of relativity. It is completed in November. After a little friction, the two men remain friendly colleagues. Einstein had created the concept of the curvature of space-time as a representation of the effects of gravity, and with Grossmann’s help, he did it with tensors. Today, general relativity remains one of the intellectual wonders of the world.

1916-18 年,德国:埃米·诺特与克莱因和希尔伯特(他们与爱因斯坦保持联系)合作,致力于加强广义相对论中能量守恒定律的解释。她在 1918 年的两个定理(现称为“诺特定理”)中找到了关键,这两个定理展示了数学对称性和物理守恒定律之间的相互联系。

1916–18, Germany: Emmy Noether works with Klein and Hilbert—who are in touch with Einstein—on tightening the interpretation of energy conservation in general relativity theory. She finds a key in her two 1918 theorems (now known as “Noether’s theorems”), which show the interconnection between mathematical symmetries and physical conservation laws.

1917 年,意大利:列维-奇维塔通过简化的比安奇恒等式展示了广义相对论与能量动量守恒定律之间的张量联系。他还使用张量来定义在曲面上求导时的“平行”概念。

1917, Italy: Levi-Civita shows the tensor connection between general relativity and conservation of energy-momentum, via the contracted Bianchi identities. He also uses tensors to define the notion of “parallel” when taking derivatives on curved surfaces.

1919 年,英国:亚瑟·爱丁顿和同事宣布了他们对光线弯曲的发现,而这需要在日全食期间进行仔细测量。这是广义相对论的首次成功新测试,爱因斯坦因此成为超级明星。他的成功使张量成为科学主流。

1919, England: Arthur Eddington and colleagues announce their findings on the bending of light, which had required careful measurements during a total solar eclipse. It is the first successful new test of general relativity, and Einstein becomes a superstar. His success puts tensors into the scientific mainstream.

20 世纪 20 年代,德国、荷兰、英国;1975 年,澳大利亚:1922 年,奥托·斯特恩和瓦尔特·格拉赫发现磁偏转电子的角动量是量子化的;不久之后,乔治·乌伦贝克和塞缪尔·古德斯米特进行了一项实验,表明斯特恩和格拉赫测量的是“自旋”角动量,而不是绕原子核旋转的电子的轨道角动量。20 世纪 20 年代中期,保罗·狄拉克在他的量子力学电子行为相对论中为自旋提供了理论支持——他使用泡利自旋矩阵来描述电子旋转,而沃尔夫冈·泡利已经证明这些矩阵的数学结构与汉密尔顿的四元数旋转完全相同。1975 年,托尼·克莱因和杰夫·奥帕特表明自旋是物理的,而不仅仅是数学类比。

1920s, Germany, Netherlands, England; 1975, Australia: In 1922, Otto Stern and Walther Gerlach find the angular momentum of electrons magnetically deflected is quantised; soon afterward, George Uhlenbeck and Samuel Goudsmit perform an experiment that suggests Stern and Gerlach had measured “spin” angular momentum, not the orbital angular momentum of electrons orbiting nuclei. In the mid-1920s Paul Dirac provides theoretical support for spin in his relativistic theory of quantum mechanical electron behavior—he uses Pauli spin matrices to describe electron rotations, and Wolfgang Pauli had shown that the maths of these has exactly the same structure as Hamilton’s quaternion rotations. In 1975, Tony Klein and Geoff Opat show that spin is physical, not just a mathematical analogy.

1924 年,荷兰:Jan Schouten 和 Dirk Struik 强调了诺特定理与广义相对论中重要的张量比安奇恒等式之间的关系。

1924, Netherlands: Jan Schouten and Dirk Struik highlight the relationship between Noether’s theorems and the tensorial Bianchi identities that are important in general relativity.

1920 年代,法国:埃利·嘉当 (Élie Cartan) 帮助发展了现代微分几何。嘉当和其他人在 20 世纪所做的这项工作,将张量(和矢量)转化为现代形式,即算符和线性映射,而不是里奇规则,该规则展示了分量在这些映射和坐标变换下如何变换。

1920s, France: Élie Cartan helps develop modern differential geometry. This work, by Cartan and others over the twentieth century, puts tensors (and vectors) into modern form, as operators and linear mappings, rather than Ricci’s rules showing how components transform under those mappings and coordinate transformations.

1960 年,美国:爱因斯坦预测的引力红移被哈佛大学的罗伯特·庞德和格伦·雷布卡通过实验探测到。(丹尼尔·波普尔在 1954 年从天文光谱中发现了不太确定的证据。)

1960, United States: The gravitational redshift predicted by Einstein is experimentally detected by Harvard University’s Robert Pound and Glen Rebka. (Less certain evidence, from astronomical spectra, had been found by Daniel Popper in 1954.)

1960 年代至 70 年代,美国:在 Gerry Salton 的带领下,向量和矩阵被用于编程搜索引擎。

1960s–70s, United States: Following the lead of Gerry Salton, vectors and matrices are used in programming search engines.

1981 年,美国:美国宇航局开始定期使用四元数来帮助引导其航天器。

1981, United States: NASA starts routinely using quaternions to help guide its spacecraft.

20 世纪 90 年代末,美国:拉里·佩奇和谢尔盖·布林在他们的 Google PageRank 算法中使用向量和矩阵。

Late 1990s, United States: Larry Page and Sergey Brin use vectors and matrices in their Google PageRank algorithm.

21 世纪初,美国:四元数用于机器人、CGI、分子动力学、手机屏幕旋转、航天器控制等领域。

Early 2000s, United States: Quaternions are used in robotics, CGI, molecular dynamics, the rotations of our mobile phone screens, spacecraft control, and much more.

2011 年,美国:美国宇航局宣布,其引力探测器 B 卫星的结果证实了爱因斯坦对时空扭曲(大地测量效应)和参考系拖拽(旋转物体拖拽空间和时间的量)的预测。

2011, United States: NASA announces that its Gravity Probe B satellite results confirm Einstein’s predictions of the warping of space-time (the geodetic effect) and frame-dragging (the amount by which a spinning object pulls space and time with it).

2015 年,国际 LIGO 合作:首次探测到广义相对论所预测的引力波。随后几年又有新的发现。

2015, international LIGO collaboration: Gravitational waves, as predicted by general relativity, are detected for the first time. New detections are revealed over subsequent years.

2017 年:国际天文学联合会在新的恒星命名系统中纳入了八十六个土著星名,以承认古代文化。

2017: The International Astronomical Union acknowledges ancient cultures by including eighty-six indigenous star names in a new starnaming system.

2018,法国:改进的测试证实了引力红移的存在(等效原理)。

2018, France: An improved test confirms the existence of gravitational redshift (equivalence principle).

2019 年,国际合作(事件视界望远镜):首次直接拍摄黑洞(阴影)图像。

2019, international collaboration (Event Horizon Telescope): The first direct image of (the shadow of ) a black hole is created.

2020 年,英国/瑞典:罗杰·彭罗斯 (Roger Penrose) 因阐明广义相对论预测黑洞存在的原因而获得诺贝尔物理学奖。

2020, England/Sweden: Roger Penrose shares the Nobel Prize for Physics, for showing why general relativity predicts the existence of black holes.

正在进行:越来越严格的测试证实了广义相对论的(张量)理论。例如,2019 年,一个团队使用 LARES 和 LAGEOS 卫星以更高的精度证实了广义相对论对参考系拖拽的预测;2021 年,一个团队使用美国宇航局的核光谱望远镜阵列和欧洲航天局的 XMM-牛顿望远镜观测了黑洞背面的 X 射线(对爱因斯坦光弯曲预测的另一次测试);2022 年,MICROSCOPE 合作项目宣布等效原理获得新的精度;2022 年,事件视界望远镜首次揭示了位于我们银河系中心的超大质量黑洞的图像;2023 年,阿塔卡马宇宙学望远镜 (ACT) 合作项目利用引力透镜绘制暗物质图像,帕克斯射电望远镜 Murriyang 发现了引力波的新证据;等等。

Ongoing: Increasingly stringent tests confirm the (tensor) theory of general relativity. For example, in 2019, a team using the LARES and LAGEOS satellites confirms to higher accuracy the general relativistic prediction of frame-dragging; in 2021, a team using NASA’s Nuclear Spectroscopic Telescope Array and the European Space Agency’s XMM-Newton Telescope observes X-rays from the back of a black hole (another test of Einstein’s light-bending prediction); in 2022, the MICROSCOPE collaboration announces new accuracy for the equivalence principle; in 2022, the Event Horizon Telescope reveals the first image of the supermassive black hole at the centre of our galaxy; in 2023, the Atacama Cosmology Telescope (ACT) collaboration uses gravitational lensing to map dark matter, and the Parkes radio telescope Murriyang finds new evidence of gravitational waves; and much more.

如今,在世界范围内,四元数、向量和张量是物理学、工程学、IT(包括人工智能、CGI 和搜索引擎)等众多领域应用的基础——事实上,几乎在所有需要精确定位空间中的物体或表示和处理信息的领域中。

Today, worldwide: Quaternions, vectors, and tensors are fundamental in a huge variety of applications in physics, engineering, IT (including AI, CGI, and search engines)—in fact, in just about everything that requires pinpointing objects in space or representing and processing information.

致谢

ACKNOWLEDGMENTS

首先,我要感谢我所生活和工作的这片土地的传统守护者——布努荣人,并向他们的过去和现在的长者表示敬意。

I want to begin by acknowledging the Traditional Custodians of the land on which I live and work—the Bunurong people—and paying my respects to their Elders past and present.

本书的诞生得益于芝加哥大学出版社无与伦比的 Joe Calamia 的洞察力和鼓励。我非常感谢 Joe,不仅因为他大胆而富有创意地建议我写一个关于向量和张量的故事,还因为他持续不断的编辑技巧和不懈的支持。与他一起工作绝对是一种荣幸。

This book owes its inception to the insight and encouragement of the incomparable Joe Calamia, at the University of Chicago Press. I’m immensely grateful to Joe, not just for his bold and creative suggestion to write a story of vectors and tensors but for his ongoing editorial skill and unflagging support. It has been an absolute pleasure to work with him.

与芝加哥团队的其他成员以及 NewSouth Publishing 的优秀团队一起工作很有趣(而且非常专业),我特别感谢 Harriet McInerney、Joumana Awad 和 Caitlin Lawless:非常感谢你们所有人!还要特别感谢 Susan Olin,她对细节的细致观察使我的语法更加流畅,故事的各个方面更加清晰,她的耐心、鼓励和幽默感使这个项目艰难的文字编辑阶段变得轻松。我还非常感谢匿名审稿人,他们为改进故事情节的历史方法和技术细节提出了善意和宝贵的建议。还要感谢 Tobiah Waldron,感谢他对数学的热爱和编制索引。

It’s been fun (and superbly professional) working with the rest of the Chicago team, too, and with the wonderful team at NewSouth Publishing, where I’m especially indebted to Harriet McInerney, Joumana Awad, and Caitlin Lawless: huge thanks to you all! Special thanks, too, to Susan Olin, whose meticulous eye for detail has smoothed out my grammar and clarified aspects of the story and whose patience, encouragement, and good humour eased the difficult copyediting stage of this project. I’m also extremely grateful to the anonymous referees for their kind and invaluable suggestions for improving both the historical approach to the storyline and the technical details. And thank you Tobiah Waldron, for loving math and preparing the index.

我还要非常感谢那些慷慨地允许我在本书中使用图片的人士和组织。在这个过程中,“陌生人的善意”确实让我精神一振,因此非常感谢剑桥大学三一学院图书馆的 Steven Archer 提供年轻的麦克斯韦;感谢佩特沃斯庄园的马克斯·埃格雷蒙特勋爵和西萨塞克斯郡档案馆的档案员阿比盖尔·哈特利提供哈里奥特的手稿;感谢都柏林高等研究院的 Michelle Tobin 和梅努斯大学汉密尔顿研究所的 David Malone 帮助我找到汉密尔顿和他的儿子,感谢爱尔兰皇家学院的 Meabdh Murphy 提供图片;感谢 Joe Calamia 提供麦克斯韦美丽的雕像;感谢詹姆斯·克拉克·麦克斯韦基金会 (JCMF) 的 Catherine Booth,她在 JCMF 的网站管理员和策展人的帮助下,在最后一刻挺身而出提供泰特的图片,继承了麦克斯韦善良的名声。 (我强烈推荐参观位于爱丁堡印度街的麦克斯韦出生地的 JCMF 博物馆。)我还要感谢剑桥大学彼得豪斯学院的 Justine Kent 的慷慨,以及苏黎世联邦理工学院图书馆的 Heike Hartmann 向我介绍了爱因斯坦、马里奇、格罗斯曼和闵可夫斯基的照片。我还非常感谢我与莫纳什大学数学学院的合作,以及莫纳什大学奇妙的图书馆。

Many thanks are also due to the people and organisations that have generously permitted me to use the images in this book. The “kindness of strangers” in this process really lifted my spirits, so huge thanks to Steven Archer, of the Trinity College Library at Cambridge University, for young Maxwell; Lord Max Egremont of Petworth House and archivist Abigail Hartley of the West Sussex Records Office, for Harriot’s manuscript; Michelle Tobin of the Dublin Institute of Advanced Studies and David Malone of the Hamilton Institute at Maynooth University, for helping me track down Hamilton and his son, and to Meabdh Murphy of the Royal Irish Academy for providing the image; to Joe Calamia for the beautiful statue of Maxwell; and to Catherine Booth of the James Clerk Maxwell Foundation ( JCMF), who, with help from the webmaster and the curator at JCMF, carried on Maxwell’s reputation for kindness by stepping in at the eleventh hour to provide the image of Tait. (I can highly recommend a visit to the JCMF Museum at Maxwell’s birthplace in India Street, Edinburgh.) I also thank Justine Kent of Peterhouse College, Cambridge, for her generosity, and Heike Hartmann of ETH-Bibliothek, Zurich, for directing me to the images of Einstein, Marić, Grossmann, and Minkowski. I’m also most grateful for my affiliation with the School of Mathematics at Monash University, and to Monash’s marvelous library.

我也向所有喜欢我作品的读者表示感谢,尤其是多年来给我写信的那些陌生人。对于一个孤独的作家来说,能感受到自己是这样一个遥远社区的一部分,是一种谦卑和美妙的感觉。对于那些离家较近的人,我特别感谢我的朋友吉娜·沃德 (Gina Ward) 和伊卡·威利斯 (Ika Willis),他们对本书前几章的支持和敏锐的反馈来得正是时候——吉娜,我们长期的友谊和持续的文学兴趣从一开始就支撑着我。也非常感谢乌尔苏拉和沃纳·泰纳特 (Werner Theinert),感谢你们对我作品的特殊友谊和宝贵支持。

I also offer my gratitude to all the readers who have enjoyed my books, especially the kind strangers who have written to me over the years. It is humbling and wonderful for a solitary writer to feel part of such a far-flung community. And to those closer to home, I especially thank my friends Gina Ward and Ika Willis, whose support and perceptive feedback on early chapters of this book came at just the right time—and Gina, our long friendship and ongoing literary interests have sustained me from the beginning. A big thank you, too, to Ursula and Werner Theinert, for your special friendship and valuable support of my work.

还要感谢 Joe Mazur、Carolyn Landon、Cheryl Hingley、Elizabeth Finkel、Erica Jolly、Margaret Harris、Vera Ray (RIP)、Penny and Molly Anggo、Bet Sibley、Catherine Watson、Phil Henshall、Anne and Phil Dempster、Peter and Anne-Marie Biram、John and Mary Mutsaers、LianeArno、Matt Stone、John 和 Helen Laing、Annie Bain、Karin Murphy-Ellis、John Di Stefano、Elizabeth 和 Ian Fraser、Ingmar Quist、Tricia Szirom、Sandra Shotlander、Susan Hawthorne、Harry Freeman、Gill Heal、Michael 和 Gail Box、Margaret Pitt 以及 John 和 Peter Snare。多年来,你们每个人都以某种特殊的方式支持我的工作 — — 从同事的支持,到在我需要时真正对我所写的内容感兴趣,到阅读(甚至购买)我的书并提供慷慨的反馈,或者理解当我全神贯注于一本书或赶着赶截止日期时,我可能会暂时不参加社交活动。

Many thanks, too, to Joe Mazur, Carolyn Landon, Cheryl Hingley, Elizabeth Finkel, Erica Jolly, Margaret Harris, Vera Ray (RIP), Penny and Molly Anggo, Bet Sibley, Catherine Watson, Phil Henshall, Anne and Phil Dempster, Peter and Anne-Marie Biram, John and Mary Mutsaers, Liane Arno, Matt Stone, John and Helen Laing, Annie Bain, Karin Murphy-Ellis, John Di Stefano, Elizabeth and Ian Fraser, Ingmar Quist, Tricia Szirom, Sandra Shotlander, Susan Hawthorne, Harry Freeman, Gill Heal, Michael and Gail Box, Margaret Pitt, and John and Peter Snare. Over the years you have each been supportive of my work in some special way—from collegial support to taking a genuine interest in what I’m writing just when I needed it, to reading (even buying) my books and offering generous feedback, or to understanding that when I’m engrossed in a book or rushing to meet deadlines I can be out of social action for a while.

最后,衷心感谢我亲爱的摩根,她无尽的支持和鼓励无与伦比。我将这本书献给你。

Finally, huge, heartfelt thanks to my darling Morgan, whose endless support and encouragement have been beyond compare. I dedicate this book to you.

笔记

NOTES

序幕

PROLOGUE

1 .威廉·罗文·汉密尔顿 (William Rowan Hamilton) 写给 PG Tait 的信,发表于 PG Tait 《四元数初等论述》第三版前言中(剑桥:剑桥大学出版社,1890 年)。

1.Letter from William Rowan Hamilton to P. G. Tait, published in the preface to the third edition of P. G. Tait, An Elementary Treatise on Quaternions (Cambridge: Cambridge University Press, 1890).

2麦克斯韦并没有使用“矢量场”这个术语,但这是他所创造的(正如他在几年后 1873 年出版的《电磁学论文》中明确指出的那样);我将在后续章节中探讨这意味着什么以及它如何影响后来的物理学家。

2.Maxwell didn’t use the term “vector field,” but that’s what he’d created (as he made clear a few years later, in his 1873 Treatise on Electricity and Magnetism); I’ll explore what this means—and how it influenced later physicists—in subsequent chapters.

3 .古代表格数学:例如,参见 Eleanor Robson 的《牛津阿什莫林博物馆的数学楔形文字板》,SCIAMVS 5(2004 年):2-65;Duncan J. Melville 的《早期美索不达米亚的计算》,《电子计算器出现之前的数学教育中的计算和计算设备》,A. Volkov 和 V. Freiman 编辑(瑞士:Springer Nature,2018 年),25-47。Melville 强调了解读此类古代文献及其预期用途的困难;Robert Middeke-Conlin 指出,关于一些美索不达米亚数学板的实用性的争论仍在继续,见《拉尔萨和巴比伦王国的运河建设数学》,《水历史》 12(2020 年):105-28。然而,丹尼尔·曼斯菲尔德 (Daniel Mansfield) 2021 年的研究为其中一些解释增添了显著的清晰度;参见他的“普林普顿 322:矩形研究”,《科学基础》 26(2021):977–1005。

3.Ancient tabular mathematics: see, e.g., Eleanor Robson, “Mathematical Cuneiform Tablets in the Ashmolean Museum, Oxford,” SCIAMVS 5 (2004): 2–65; Duncan J. Melville, “Computation in Early Mesopotamia,” in Computations and Computing Devices in Mathematics Education before the Advent of Electronic Calculators, ed. A. Volkov and V. Freiman (Switzerland: Springer Nature, 2018), 25–47. Melville highlights the difficulties in interpreting such ancient documents and their intended uses; and Robert Middeke-Conlin points out that the debate over the utility of some of the Mesopotamian mathematical tablets is ongoing, in “The Mathematics of Canal Construction in the Kingdoms of Larsa and Babylon,” Water History 12 (2020): 105–28. Daniel Mansfield’s 2021 study, however, adds significant clarity to some of these interpretations; see his “Plimpton 322: A Study of Rectangles,” Foundations of Science 26 (2021): 977–1005.

4 .我很感激曼斯菲尔德在《普林普顿 322》中提供的有关美索不达米亚土地使用和测量以及普林普顿角色的信息。曼斯菲尔德发现了 Si427 的毕达哥拉斯性质,他还对普林普顿 322 给出了新的见解以及有关美索不达米亚乘法表的详细信息。然而,关于他对美索不达米亚三角学的主张,请参见伊芙琳·兰姆的《不要被巴比伦三角学的炒作所蒙骗》,《科学美国人》(博客),2017 年 8 月 29 日。

4.I’m indebted for the information on Mesopotamian land use and surveying, and on Plimpton’s role, to Mansfield, “Plimpton 322.” Mansfield discovered the Pythagorean nature of Si427, and he also gives new insights on Plimpton 322 as well as details about Mesopotamian multiplication tables. On his claims for Mesopotamian trigonometry, however, see, e.g., Evelyn Lamb, “Don’t Fall for Babylonian Trigonometry Hype,” Scientific American (blog), August 29, 2017.

5澳大利亚土著天文学:Duane Hamacher,《天空的故事:土著知识中的天文学》,《对话》 ,2014 年 12 月 1 日;Ray P. Norris、Cilla Norris、Duane W. Hamacher 和 Reg Abrahams,《Wurdi Youang:可能具有太阳迹象的澳大利亚土著石雕》,《岩石艺术研究》 30 卷,第 1 期(2013 年):55–65 页。

5.Australian indigenous astronomy: Duane Hamacher, “Stories from the Sky: Astronomy in Indigenous Knowledge,” The Conversation, December 1, 2014; Ray P. Norris, Cilla Norris, Duane W. Hamacher, and Reg Abrahams, “Wurdi Youang: An Australian Aboriginal Stone Arrangement with Possible Solar Indications,” Rock Art Research 30, no. 1 (2013): 55–65.

6 .托勒密的一些杰出资料现在大多已失传,其中包括尼多斯的欧多克索斯,他或许是第一个开始天文学几何建模过程的人,也是我们将在第 2 章中介绍的原始微积分穷举法的先驱;昔兰尼的埃拉托色尼,他显然使用了一种基本的纬度和经度,并且由于使用了影棒和简单的测量杆,他极其准确地推断出了地球的大小;尼西亚的喜帕恰斯,他似乎是第一个系统地使用 360° 圆来精确地几何表示行星运动的人,他的数学和天文学水平非常高,以至于发现了春分点进动;以及佩尔加的阿波罗尼乌斯,他在圆锥曲线的数学分析中使用了一种事后坐标系,托勒密直接以他的本轮和行星运动偏心模型为基础。

6.Some of Ptolemy’s brilliant but now mostly lost sources are Eudoxus of Cnidus, perhaps the first to begin the process of geometric modeling in astronomy, and who pioneered the protocalculus method of exhaustion that we’ll meet in chap. 2; Eratosthenes of Cyrene, who apparently used a basic kind of latitude and longitude, and who deduced the size of the Earth extraordinarily accurately, given he used a shadow stick and simple measuring rods; Hipparchus of Nicaea, who seems to have been the first to systematically use a 360° circle to make precise geometrical representations of planetary motion, and whose maths and astronomy were so good that he discovered the precession of the equinoxes; and Apollonius of Perga, who used a kind of post-hoc coordinate system in his mathematical analysis of conics, and whose epicycle and eccentric models of planetary motion Ptolemy built directly upon.

7例如,如果你以每小时 35 英里的速度向东北行驶,你的矢量将指向与轴线成 45° 的方向,但它的分量将是这样的,当你使用勾股定理将它们相加时,量级仍然是 35。所以,它将是矢量三十五2三十五2

7.For example, if you were traveling, say, northeast at 35 mph, your vector would point in the direction at 45° to the axes, but its components would be such that when you add them using Pythagoras’s theorem, the magnitude is still 35. So, it would be the vector 352,352.

向量三十五2三十五2从图 0.2中可以看出,在计算所需组件时四十五°=余弦四十五°=12;如果你对三角函数不太熟悉,可以先看一下图 3.4,看看为什么我们需要 sin 45° 和 cos 45° 作为垂直和水平分量。然后,利用毕达哥拉斯定理,你得到三十五22+三十五22=三十五2,所以矢量是三十五2三十五2震级为 35 级。

The vector 352,352 is found from fig. 0.2, and the fact that when working out the components you need sin45°=cos45°=12; if you’re rusty on trigonometry, you can take a look ahead at fig. 3.4 to see why we need sin 45° and cos 45° for the vertical and horizontal components. Then, using Pythagoras’s theorem, you have 3522+3522=352, so the vector is 352,352 and the magnitude is 35.

8四速度分量的定义与普通速度类似,都是(时空)坐标相对于(本征)时间的导数。本征时间是可直接测量的“时钟时间”,即相对于观察者静止的时钟。

8.The components of four-velocity are defined by analogy with ordinary velocity, as the derivatives of the (space-time) coordinates with respect to (proper) time. The proper time is the directly measurable “clock time,” that is, from a clock at rest with respect to the observer.

9疯帽子戏仿猜想指的是汉密尔顿的一种新乘法类型(非交换性,我们将在第 1 章和第 4 章中看到)的结果;它是由维多利亚文学专家梅兰妮·贝利在《爱丽丝的代数历险记:仙境解开》中提出的,新科学家,2009 年 12 月 16 日。迈克尔·布鲁克斯在他的《更多艺术》(墨尔本:Scribe,2021 年)中跟进了贝利的故事,第 194-96 页。

9.The Mad Hatter parody conjecture refers to a consequence of one of Hamilton’s new types of multiplication (noncommutativity, which we’ll see in chaps. 1 and 4); it was suggested by Victorian literature expert Melanie Bayley, “Alice’s Adventures in Algebra: Wonderland Solved,” New Scientist, December 16, 2009. Michael Brooks followed up the story with Bayley in his The Art of More (Melbourne: Scribe, 2021), 194–96.

布鲁克斯和贝利将道奇森描绘成一个保守的、平庸的数学家(另见迈克尔·迪肯等,“刘易斯·卡罗尔——数学家?” Function 18,no. 1 [1994 年 2 月]:10–18)——一个不可能理解汉密尔顿著作的人。这很可能是真的。然而,道奇森/卡罗尔如今在数学上的名声并非建立在他生前发表的著作上,而是建立在他死后发现的关于投票方法和符号逻辑的手稿上(在其中他使用了荒谬的命题和符号代数来教授逻辑规则);参见弗朗辛·阿贝莱斯​​,“逻辑和刘易斯·卡罗尔”,自然527(2015 年 11 月 19 日):302–4;阿米鲁什·莫克特菲,“当你可以把事情复杂化时,为什么要把事情简单化?对刘易斯·卡罗尔符号逻辑的欣赏”,《Logica Universalis》15(2021):359–79——若想了解卡罗尔的逻辑谜题,请参见夏威夷大学网站http://math.hawaii.edu/~hile/math100/logice.htm

Brooks and Bayley paint Dodgson as a conservative, mediocre mathematician (see also, e.g., Michael Deakin, “Lewis Carroll—Mathematician?,” Function 18, no. 1 [February 1994]: 10–18)—one who couldn’t have understood Hamilton’s work. This may well be true. Dodgson/Carroll’s mathematical fame today, however, rests not on work published in his lifetime but on posthumously discovered later manuscripts on methods of voting and on symbolic logic (in which he used absurd propositions and symbolic algebra to teach the rules of logic); cf. Francine Abeles, “Logic and Lewis Carroll,” Nature 527 (November 19, 2015): 302–4; Amirouche Moktefi, “Why Make Things Simple When You Can Make Them Complicated? An Appreciation of Lewis Carroll’s Symbolic Logic,” Logica Universalis 15 (2021): 359–79—and, for a taste of Carroll’s logical puzzles, see, e.g., the University of Hawaii website http://math.hawaii.edu/~hile/math100/logice.htm.

但请注意,在卡罗尔的《符号逻辑》(纽约:多佛,1958 年;最初出版于 1897 年,比艾丽丝晚三十年)中,他建立了符号逻辑的交换代数(例如 35、70):卡罗尔的方程是关于逻辑的,而不是代数的,但有趣的是他选择强调“交换性”——也许他确实对汉密尔顿的非交换乘法有疑问。或者他只是想将他的符号逻辑命题与矢量积区分开来。

Note, however, that in Carroll’s Symbolic Logic (New York: Dover, 1958; originally published 1897, three decades after Alice), he sets up a commutative algebra of symbolic logic (e.g., 35, 70): Carroll’s equation is about logic, not algebra, but it is interesting that he chose to emphasise “commutativity”— perhaps he did have a problem with Hamilton’s noncommutative multiplication. Or perhaps he simply wanted to distinguish his symbolic logical propositions from vector products.

第一章

CHAPTER 1

1 .在他的《四元数讲座》序言中,汉密尔顿说他之所以研究四元数是因为他想“将计算几何联系起来”,并将这些计算“从平面转移到空间”。后来他展示了如何进行三维旋转;参见《四元数讲座》(都柏林:霍奇斯和史密斯出版社,伦敦:惠特克出版社和剑桥:麦克米伦出版社,1853 年),第 269 页(第 282 页)。

1.In the preface to his Lectures on Quaternions, Hamilton said he was led to quaternions because he wanted to “connect calculation with geometry,” and to move these calculations “from the plane to space.” Later he shows how to do 3-D rotations; see Lectures on Quaternions (Dublin: Hodges and Smith, London: Whittaker, and Cambridge: Macmillan, 1853), 269 (art. 282).

2梅兰妮·贝利(《爱丽丝的代数历险记》作者)指出,这个例子和其他例子表明卡罗尔是在戏仿汉密尔顿。但卡罗尔可能只是在区分逻辑规则和汉密尔顿的代数规则。

2.Melanie Bayley (“Alice’s Adventures in Algebra”) pointed out this and other examples that suggest, to her, that Carroll was parodying Hamilton. But it’s possible that Carroll might simply have been distinguishing the rules of logic from those of Hamilton’s algebra.

3 .汉密尔顿的电类比来自写给他儿子阿奇博尔德的一封信(引自 Michael J. Crowe 的《矢量分析史》 [印第安纳州诺特丹:诺特丹大学出版社,1967 年],第 29-30 页)。他在 1858 年的一封信中给了 PG Tait 类似的描述:https://www.tcd.ie/library/manuscripts/blog/tag/moon-landing/。纪念艺术装置由 Emma Ray 创作;请参阅共同委托这项工作的爱尔兰皇家学院的网站,特别是准备和安装的 YouTube 视频,网址为https://www.google.com/search?client=safari&rls=en&q=Commemorative+art+installation+Hamilton+Broombridge+Luas&ie=UTF-8&oe=UTF-8#fpstate=ive&vld=cid:a9bfbdde,vid:1nQct3p3184

3.Hamilton’s electrical analogy is from a letter to his son Archibald (quoted in Michael J. Crowe, A History of Vector Analysis [Notre Dame, IN: University of Notre Dame Press, 1967], 29–30). He gave a similar description to P. G. Tait in an 1858 letter: https://www.tcd.ie/library/manuscripts/blog/tag/moon-landing/. The commemorative art installation is by Emma Ray; see the website of the Royal Irish Academy, which cocommissioned the work, and especially the YouTube video of the preparation and installation, at https://www.google.com/search?client=safari&rls=en&q=Commemorative+art+installation+Hamilton+Broombridge+Luas&ie=UTF-8&oe=UTF-8#fpstate=ive&vld=cid:a9bfbdde,vid:1nQct3p3184.

4 .在参观都柏林圣三一学院的老图书馆时,阿姆斯特朗在汉密尔顿的大理石半身像旁停留,向导游解释四元数如何帮助航天器导航:Estelle Gittins,2019 年 7 月 19 日,https://www.tcd.ie/library/manuscripts/blog/tag/moon-landing/

4.While visiting the Old Library at Trinity College, Dublin, Armstrong paused beside a marble bust of Hamilton and explained to his guide how quaternions help spacecraft navigation: Estelle Gittins, July 19, 2019, https://www.tcd.ie/library/manuscripts/blog/tag/moon-landing/.

5我改编了 TL Heath 所著《欧几里得几何原本译本》(剑桥:剑桥大学出版社,1925 年)中可能出自毕达哥拉斯的一张图表,该图表转载自 John Stillwell 所著《数学及其历史》(纽约:Springer-Verlag,1989 年),第 7 页。有关欧几里得更复杂的证明,请参阅 Carl Boyer 所著《数学史》,修订版由 Uta Merzbach 所著(纽约:John Wiley and Sons,1991 年),第 108 页。

5.I have adapted a diagram—possibly attributed to Pythagoras—given in T. L. Heath, Translation of Euclid’s Elements (Cambridge: Cambridge University Press, 1925) reproduced in John Stillwell, Mathematics and Its History (New York: Springer-Verlag, 1989), 7. For Euclid’s more sophisticated proof, see Carl Boyer, A History of Mathematics, rev. Uta Merzbach (New York: John Wiley and Sons, 1991), 108.

6 .有不同的版本,但请参阅例如国会图书馆,https://www.loc.gov/item/2021666184/

6.There are different versions, but see, e.g., Library of Congress, https://www.loc.gov/item/2021666184/.

7现代数学家倾向于将i定义为x 2 + 1 = 0的(主)解,而不是将其指定为1;换句话说,i通常以其平方来定义,即i 2 = −1,而不是平方根。这是因为后者可能导致如下难题:

7.Modern mathematicians tend to prefer defining i as the (principal) solution to x2 + 1 = 0, rather than specifying it as 1; in other words, i is usually defined in terms of its square, i2 = −1, rather than as a square root. That’s because the latter can lead to conundrums such as this:

1=×=1×111=1=±1

1=i×i=1×111=1=±1,

如果取正根,则有 −1 = 1,这显然是错误的!

and if you take the positive root, you have −1 = 1, which is clearly wrong!

8笛卡尔论虚数:Brian E. Blank,《书评:Paul Nahin 的《一个虚构的故事》》,AMS 通告(1999 年 11 月):1233。

8.Descartes on imaginary numbers: Brian E. Blank, “Book Review: An Imaginary Tale by Paul Nahin,” Notices of the AMS (November 1999): 1233.

9博耶在《数学史》第 229 页引用了 Al-Khwārizmī 的论述;他用几何方法完成平方,第 231 页。Al -jabr的翻译……,以及几何例如:Raymond Flood 和 Robin Wilson,《伟大的数学家》(伦敦:Arcturus,2011),46–47。

9.Al-Khwārizmī quoted in Boyer, History of Mathematics, 229; his geometrical way of completing the square, 231. Translation of Al-jabr … , and geometric example: Raymond Flood and Robin Wilson, The Great Mathematicians (London: Arcturus, 2011), 46–47.

10有关哈里奥特非凡的一生和工作的更多信息,请参阅我的《托马斯·哈里奥特:科学人生》(纽约:牛津大学出版社,2019 年)及其参考文献。请注意,他的遗作《实践》是由他的朋友编写的,但显然他们不是像他那样优秀的数学家:他的论文提供了比他们发表的论文更复杂的工作,包括虚数的使用。

10.For more about Harriot’s extraordinary life and work, see my Thomas Harriot: A Life in Science (New York: Oxford University Press, 2019), and the references therein. Note that his posthumous book, Praxis, was put together by his friends, but evidently they were not such good mathematicians as he: his papers offer more sophisticated work than that which they published, including the use of imaginary numbers.

11有关代数符号演变的迷人历史,请参阅 Joseph Mazur 的《启蒙符号:数学符号简史及其隐藏的力量》(新泽西州普林斯顿:普林斯顿大学出版社,2014 年)。

11.For a fascinating history of the evolution of algebraic symbolism, see Joseph Mazur, Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers (Princeton, NJ: Princeton University Press, 2014).

12除了狭义相对论和E = mc 2论文外,爱因斯坦在 1905 年还发表了两篇关于布朗运动和分子大小的重要论文,以及一篇关于光量子理论的开创性论文。有关介绍性概述,请参阅我的《年轻的爱因斯坦和 E = mc 2的故事》(悉尼:Ligature,2014 年)。

12.In addition to his special relativity and E = mc2 papers, in 1905 Einstein also published two important papers on Brownian motion and the size of molecules as well as a pioneering paper on the quantum theory of light. For an introductory overview, see my Young Einstein and the Story of E = mc2 (Sydney: Ligature, 2014).

13这个问题来自 CBS 43 号泥板,Eleanor Robson 在其著作《费城楔形文字数学第一部分:问题与计算》SCIAMVS 1(2000):11–48 中翻译了这个问题;为了便于说明,我将问题的右侧改为 21 — 泥板(39 号泥板)上的数字是 41,但正如 Robson 所说(42),泥板上的符号并不完全清晰,而且经破译后,无法得到当时使用的那种简单整数或两位数(六十进制)解。我对古巴比伦几何方法的配平方图改编自第 42 页

13.This problem is from tablet CBS 43, as translated in Eleanor Robson, “Mathematical Cuneiform Tablets in Philadelphia, Part I: Problems and Calculations,” SCIAMVS 1 (2000): 11–48; for my illustrative purpose, I have changed the right-hand side of the problem to 21—the tablet (shown on 39) has 41, but as Robson says (42), the symbols are not entirely clear on the tablet, and as deciphered do not yield the kind of simple integer or two-place (in the sexagesimal system) solution used at the time. My diagram of the Old Babylonian geometrical method of completing the square is adapted from p. 42.

14运河等:Robert Middeke-Conlin,《拉尔萨和巴比伦王国的运河建设数学》,《水史》 12 (2020):105–28。

14.Canals, etc.: Robert Middeke-Conlin, “The Mathematics of Canal Construction in the Kingdoms of Larsa and Babyon,” Water History 12 (2020): 105–28.

15请注意,早期的数学家(包括 12 世纪早期的波斯传奇诗人奥玛·海亚姆,他也是一位数学家)已经找到了一种纯几何的方式,通过两条曲线的交点来解决一些具有正根的三次方程:博耶的《数学史》,241;弗勒德和威尔逊的《伟大的数学家》,49。关于 al-Ṭūsī,请参阅 JJ O'Connor 和 EF Robertson 为他撰写的 MacTutor 条目,网址为https://mathsh isstory.st-andrews.ac.uk/Biographies/Al-Tusi_Sharaf/

15.Note that earlier mathematicians—including the legendary early twelfthcentury Persian poet Omar Khayyam, who was also a mathematician—had found a purely geometrical way of solving some cubic equations with positive roots via the intersection of two curves: Boyer, History of Mathematics, 241; Flood and Wilson, The Great Mathematicians, 49. On al-Ṭūsī, see J. J. O’Connor and E. F. Robertson’s MacTutor entry for him, at https://mathsh istory.st-andrews.ac.uk/Biographies/Al-Tusi_Sharaf/.

16卡尔达诺用于求解x 3 = cx + d形式的方程的底层算法(基于塔塔利亚算法)如下:选择新变量u、v,并设置x = u + v,uv = c/3。将这些代入原始方程,您将得到u 3 + v 3 = d;消除v ,这变成了u 3的一个二次方程,可以用二次公式求解。将u 3的解代入u 3 + v 3 = d中,解出v 3 。对u 3v 3取立方根,求出u 和 v,从而得到x = u + v。这个公式非常巧妙,而且都没有使用现代符号系统,这样更容易跟踪你的思维过程。我给出的例子x 3 = 6 x + 40 ,以及卡尔达诺求解它的算法——以及他对立方体的几何完成——都在他的《大艺斯》第 12 章中,该书重印于 R. Laubenbacher 和 D. Pengelley 的《代数:寻找一个难以捉摸的公式》中, 《数学探险》 ,本科生数学教材(纽约:Springer,1999 年),230 页;https://doi.org/10.1007/978-1-4612-0523-4_5

16.Cardano’s underlying algorithm (based on Tartaglia’s) for solving an equation of the form x3 = cx + d is this: choose new variables u, v and set x = u + v, uv = c/3. Put these into the original equation, and you’ll get u3 + v3 = d; eliminate v and this becomes a quadratic equation in u3, which can be solved using the quadratic formula. Put this solution for u3 into u3 + v3 = d and solve for v3. Take the cube roots of u3 and v3 to find u, v, and hence x = u + v. It’s ingenious, and all created without the modern symbolism that makes it easier to keep track of your thought processes. The example I gave, x3 = 6x + 40, and Cardano’s algorithm for solving it—together with his geometric completion of the cube—is in chap. 12 of his Ars Magna, reprinted in R. Laubenbacher and D. Pengelley, “Algebra: The Search for an Elusive Formula,” in Mathematical Expeditions, Undergraduate Texts in Mathematics (New York: Springer, 1999), 230; https://doi.org/10.1007/978-1-4612-0523-4_5.

17例如,薛定谔方程描述了光子、电子和其他亚原子粒子等基本粒子的动力学,其中就包含i。电磁波也更容易用复数形式进行数学处理,因此i是各种现代技术的幕后推手。

17.For instance, Schrödinger’s equation describes the dynamics of fundamental particles such as photons, electrons, and other subatomic particles—and it contains i. Electromagnetic waves, too, are easier to handle mathematically using the complex form, so i is behind all sorts of modern technology.

18Wallis 论哈里奥特:引自 Jacqueline Stedall 的《光荣被剥夺:托马斯·哈里奥特及其代数的死后不幸》,《精确科学史档案》第 54 卷,第 6 期(2000 年 6 月):第 490 页。哈里奥特是第一个用代数(符号)解三次方程的人:伟大的数学家拉格朗日首先提出了这一观察;参见 Seltman 的《哈里奥特的代数:声誉与现实》,载于托马斯·哈里奥特,第 1 卷,《伊丽莎白时代的科学家》,罗伯特·福克斯主编(奥尔德肖特:阿什盖特,2000 年),第 185 页。

18.Wallis on Harriot: quoted in Jacqueline Stedall, “Rob’d of Glories: The Posthumous Misfortunes of Thomas Harriot and His Algebra,” Archive for History of Exact Sciences 54, no. 6 ( June 2000): 490. Harriot first to algebraically (symbolically) solve cubics: The great mathematician Lagrange first made this observation; see Seltman, “Harriot’s Algebra: Reputation and Reality,” in Thomas Harriot, vol. 1, An Elizabethan Man of Science, ed. Robert Fox (Aldershot: Ashgate, 2000), 185.

19类似地,二次方程有两个解,四次方程有四个解,等等。德国数学家 Peter Roth 大约在同一时间提出了次数和解的数量之间的联系(在他 1608 年的《哲学算术》中),但他没有以符号形式写出他的方程式,也没有探究复数根。哈里奥特的“因子”构造暗示的“代数基本定理”的严格证明出现在 200 年后——哈里奥特本人并没有声称这么多。他使用因子和符号获得复数解的一个例子可以在例如英国图书馆手稿 6783,第 157、156 页中找到。

19.Similarly, a quadratic equation has two solutions, a quartic has four solutions, and so on. The German mathematician Peter Roth suggested this link between degree and number of solutions at around the same time (in his 1608 Arithmetica Philosphica), but he did not write his equations symbolically or explore complex roots. A rigorous proof of the “fundamental theorem of algebra” suggested by Harriot’s “factor” construction came two hundred years later—Harriot himself didn’t claim as much. An example of his use of factors and symbols to get complex solutions is found in, e.g., British Library Manuscript 6783, fols. 157, 156.

20按照欧拉(或图 3.4 和 3.6 以及第 3 章中的相关讨论),可以将复数a + ib写为r (cos θ + i sin θ) = re i θ,其中r=一个2+b2θ 可由反余弦和反正弦得出。根据棣莫弗定理(或简单地根据指数定律),该数的立方根为rθ3=r1/3θ+2π3其中k = 0, 1, 2 给出三个不同的根。将其应用于2+113+2113=r1/3θ+2π3+r1/3θ2π3=2r1/3余弦θ+2π3,你得到了 Cardano 的三个解决方案方程,=42+323。有点繁琐,但所有步骤只用到高中或大学一年级的数学。

20.Following Euler (or figs. 3.4 and 3.6 and related discussion in chap. 3), you can write a complex number a + ib as r(cos θ + i sin θ) = reiθ, where r=a2+b2 and θ is found from the inverse cosine and sine accordingly. From De Moivre’s theorem (or simply from the index laws), the cube root of this number is reiθ3=r1/3eiθ+2kπ3, where k = 0, 1, 2 gives the three different roots. Applying this to 2+11i3+211i3=r1/3eiθ+2kπ3+r1/3eiθ2kπ3=2r1/3cosθ+2kπ3, you get the three solutions of Cardano’s equation, x=4,2+3,23. It is a bit fiddly, but all the steps use only senior high school or freshman university maths.

21哈里奥特的引文:大英图书馆附加手稿 6783 fol. 186。另请参阅 Jacqueline Stedall,《托马斯·哈里奥特对弗朗索瓦·韦达论文集的注释》,《精确科学档案》第 62 卷,第 2 期(2008 年 3 月):179–200 页。

21.Harriot’s quotation: British Library Additional Manuscript 6783 fol. 186. See also Jacqueline Stedall, “Notes Made by Thomas Harriot on the Treatises of François Viète,” Archive for Exact Sciences 62, no. 2 (March 2008): 179–200.

22Seltman 的引文摘自她的《Harriot's Algebra》,收录于Thomas Harriot 的第一卷《一位伊丽莎白时代的科学家》,由 Robert Fox 编辑(Aldershot: Ashgate, 2000),第 184 页,重点是我加的。关于 Harriot 对复数和负数解的使用,Seltman 细致地分析了 Harriot 的手稿相对于他死后出版的《Artis Analyticae Praxis》的优越性,后者是由他那些能力较差的朋友根据他们对 Harriot 论文的理解整理而成的。这包括他们拒绝 Harriot 对虚数和负数的使用。

22.Seltman’s quotation is in her “Harriot’s Algebra,” in Thomas Harriot, vol. 1, An Elizabethan Man of Science, ed. Robert Fox (Aldershot: Ashgate, 2000), 184, my emphasis. Regarding Harriot’s use of complex and negative solutions, Seltman gives a fine analysis of the superiority of Harriot’s manuscripts to the posthumously published Artis Analyticae Praxis, which was put together by his less capable friends, on the basis of what they understood from his papers. This includes their rejection of Harriot’s use of imaginary and negative numbers.

第二章

CHAPTER 2

1 .声波是气压变化的结果,鹅卵石的冲击波会穿过池塘中的水,但来自太阳的光会穿过空旷的空间。那么光波中可能波动的是什么呢?人们早就假设存在神秘而不可探测的“以太”,但麦克斯韦最终给出了答案(第 6 章)。玛丽·萨默维尔的回忆录收录在她的回忆录《玛丽·萨默维尔从早年到老年的个人回忆录》中,由她的女儿玛莎·查特斯·萨默维尔编辑(伦敦:约翰·默里,1873 年),第 132 页。

1.Sound waves are changes of air pressure, and the pebbles’ shock waves travel through the water in the pond—but light travels from the sun through empty space. So what could possibly be rippling in a light wave? The mysterious, undetectable “ether” had long been postulated, but Maxwell would eventually provide the answer (chap. 6). Mary Somerville’s recollection is in her memoir, Personal Recollections from Early Life to Old Age of Mary Somerville, edited by her daughter Martha Charters Somerville (London: John Murray, 1873), 132.

2光镊利用激光束的辐射压来移动微小颗粒,这是另一个数学预测的例子:麦克斯韦从数学上预测了辐射压的存在,并在三十年后的 1901 年通过实验证实了这一点。麦克斯韦论辐射压:《电磁论》(牛津:克拉伦登出版社,1873 年,第 3 版(1891 年)1954 年由多佛重印),2:440–441(图 792–93)。

2.Optical tweezers use the laser-beams’ radiation pressure to move the tiny particles, and this is another example of a mathematical prediction: it was Maxwell who mathematically predicted the existence of radiation pressure, which was experimentally confirmed three decades later, in 1901. Maxwell on radiation pressure: Treatise on Electricity and Magnetism (Oxford: Clarendon Press, 1873, 3rd edition (1891) reprinted in 1954 by Dover), 2:440–41 (arts. 792–93).

3 .阿赫梅斯撰写了现在被称为莱因德纸莎草纸的文献,该文献以 19 世纪 50 年代在埃及购买它的收藏家的名字命名;现在该文献藏于大英博物馆。

3.Ahmes wrote what is now known as the Rhind papyrus, after the collector who bought it in Egypt in the 1850s; it’s now in the British Museum.

4 .关于牛顿:理查德·S·韦斯特福尔,《永不停息:艾萨克·牛顿传记》(剑桥:剑桥大学出版社,1980 年);韦斯特福尔还在《大英百科全书》中撰写了有关牛顿的条目。

4.On Newton: Richard S. Westfall, Never at Rest: A Biography of Isaac Newton (Cambridge: Cambridge University Press, 1980); Westfall also wrote the entry on Newton in the Encylopaedia Britannica.

5对莱布尼茨的描述来自于菲利普·P·维纳 (Philip P. Wiener) 主编的《莱布尼茨选集》 (纽约:Charles Scribner's Sons,1951 年) 导言的第一页。

5.The description of Leibniz is from the first page of the introduction to Philip P. Wiener, ed., Leibniz Selections (New York: Charles Scribner’s Sons, 1951).

6 .定义无穷小量和极限的尝试:莱布尼茨:“微分小于任何给定量”,以及“如果人们宁愿拒绝无穷小量,那么可以假设它们小到人们认为必要的程度,以便它们无法比较,产生的误差不会造成任何后果,或者小于任何给定的量级。”牛顿:“数量和数量比率在任何有限时间内不断收敛到相等,并且在这段时间结束之前彼此接近程度超过任何给定的差异,最终将变得相等。”

6.Attempts at defining infinitesimals and limits: Leibniz: “A differential is less than any given quantity,” and “If one preferred to reject infinitesimally small quantities, it was possible instead to assume them to be as small as one judges necessary in order that they should be incomparable and the error produced should be of no consequence, or less than any given magnitude.” Newton: “Quantities, and the ratios of quantities, which in any finite time converge continually to equality, and before the end of that time approach nearer to each other than by any given difference, become ultimately equal.”

现代定义:在合适的域内定义f ( x ),αf=大号如果给定任意数 ε > 0,我们都可以找到一个数 δ > 0,使得f ( x ) 满足L − ε < f ( x ) < L + ε(只要a − δ < x < a + δ)。对于x趋近于无穷大的极限,我们有f=大号如果对于任何数 ε > 0,我们都可以找到一个数M,使得当n > M时, L − ε < f ( x ) < L + ε 。此定义源自奥古斯丁·路易·柯西和卡尔·魏尔斯特拉斯等人在牛顿尝试定义极限两个世纪后所做的工作。

Modern definition: Defining f(x) in a suitable domain, limxαfx=L if, given any number ε > 0, we can find a number δ > 0 such that f(x) satisfies L − ε < f(x) < L + ε whenever a − δ < x < a + δ. For limits where x approaches infinity, we have limxfx=L if for any number ε > 0 we can find a number M such that L − ε < f(x) < L + ε when n > M. This definition arises from work by the likes of Augustin Louis Cauchy and Karl Weierstrass two centuries after Newton’s attempt at defining a limit.

7沃利斯在他的《无限算术》中对哈里奥特(以及奥特雷德和笛卡尔)表示了感谢——引自博耶的《微积分史及其概念发展》(纽约:多佛,1959 年),第 170 页;另见第 168–69 页。有关沃利斯对哈里奥特的感激之情和他异常钦佩之情的更多细节,请参阅 Stedall 的《荣耀被剥夺》,第 481–90 页。牛顿首次发表的微积分著作是在《自然哲学的数学原理》第二卷。虽然他的大部分证明都是几何形式的,但有时他也会用代数符号来给出微积分算法,例如在第二卷第二节引理 2 中。当他给出几何图表时,他有时使用符号代数来解释算法或构造,如在第二卷第 10 条命题第 3 题中。甚至在牛顿第一卷中,他也会用代数符号来解释算法或构造。 1,微积分概念通常在相关的几何构造中清晰可见(例如他对命题 39 的证明)。然而,在他关于微积分的第一份手稿中,他使用了代数而不是几何。

7.Wallis acknowledged Harriot (and also Oughtred and Descartes) in his Arithmetica Infinitorum—quoted in Boyer, The History of Calculus and Its Conceptual Development (New York: Dover, 1959), 170; see also 168–69. For more details on Wallis’s debt to and exceptionally informed admiration of Harriot, see Stedall, “Rob’d of Glories,” 481–90. Newton’s first published account of his calculus was in Principia, bk. 2. Although he gave most of his proofs geometrically, he sometimes gave the algorithms of calculus in terms of algebraic symbolism, e.g., in bk. 2, sec. 2, lemma 2. When he gave geometrical diagrams, he sometimes used symbolic algebra to explain the algorithm or construction, as in bk. 2, prop. 10, problem 3. Even in bk. 1, calculus concepts are often clearly evident in the relevant geometrical constructions (such as his proof of prop. 39). In his first manuscripts on calculus, however, he’d used algebra rather than geometry.

8沃利斯与费马等人:Jacqueline Stedall,《约翰·沃利斯与法国人:他与费马、帕斯卡、杜劳伦和笛卡尔的争吵》,《数学史》第 39 卷(2012 年):第 265-79 页。笛卡尔和哈里奥特:沃利斯的说法有些夸张,但并非他独创,因为人们不确定笛卡尔在写出著名的《几何学》之前是否看过哈里奥特的《实践》;当然,独立的共同发现并不罕见,笛卡尔比哈里奥特走得更远,但笛卡尔对其资料来源含糊其辞是出了名的,甚至他的同胞、维达的编辑让·博格兰也指出了笛卡尔的作品与哈里奥特的作品有相似之处。见Stedall,《Rob'd of Glories》,第 488–89 页,Jacqueline Stedall,《Reconstructing Thomas Harriot's Treatise on Equations》,载于Thomas Harriot,第 2 卷,《Mathematics, Exploration, and Natural Philosophy in Early Modern England》,Robert Fox 主编(法纳姆,萨里:阿什盖特出版社,2012 年),第 62 页,以及 Carl Boyer,《彩虹:从神话到数学》(新泽西州普林斯顿:普林斯顿大学出版社,1987 年),第 203 页,第 211 页。

8.Wallis vs. Fermat et al.: Jacqueline Stedall, “John Wallis and the French: His Quarrels with Fermat, Pascal, Dulaurens, and Descartes,” Historia Mathematica 39 (2012): 265–79. Descartes and Harriot: Wallis’s claims were exaggerated, but they were not original to him for there is some uncertainty about whether or not Descartes had seen Harriot’s Praxis before he wrote his famous La géometrie; of course independent codiscovery is not uncommon, and Descartes went much further than Harriot, but Descartes was notoriously vague about his sources, and even his compatriot, Viète’s editor Jean Beaugrand, noted similarities between Descartes’s work and Harriot’s. See Stedall, “Rob’d of Glories,” 488–89, and Jacqueline Stedall, “Reconstructing Thomas Harriot’s Treatise on Equations,” in Thomas Harriot, vol. 2, Mathematics, Exploration, and Natural Philosophy in Early Modern England, ed. Robert Fox (Farnham, Surrey: Ashgate, 2012), 62, and also Carl Boyer, The Rainbow: From Myth to Mathematics (Princeton, NJ: Princeton University Press, 1987), 203, 211.

9沃利斯:摘自约翰·斯蒂尔威尔(John Stillwell)的传记,《数学及其历史》(纽约:Springer-Verlag,1989 年),110–12。

9.Wallis: excerpt from his biography in John Stillwell, Mathematics and Its History (New York: Springer-Verlag, 1989), 110–12.

10沃利斯的政治命运:请参阅博德利图书馆对 J. Wallis 的《书信及其他文件集,MS e Mus. 203》的描述,网址为https://archives.bodleian.ox.ac.uk/repositories/2/resources/5805;以及 JJ O'Connor 和 EF Robertson 的描述,网址为 https://mathshistory.st-andrews.ac.uk/Biographies/Wallis/

10.Wallis’s political fortunes: See the Bodleian Library’s description of J. Wallis, A Collection of Letters and Other Papers, MS e Mus. 203, at https://archives.bodleian.ox.ac.uk/repositories/2/resources/5805; and J. J. O’Connor and E. F. Robertson, https://mathshistory.st-andrews.ac.uk/Biographies/Wallis/.

11爱因斯坦对牛顿的评价来自于他的《观念与观点》(1954年;纽约:三河出版社,1982年),第254-55页。

11.Einstein’s assessment of Newton is from his Ideas and Opinions (1954; New York: Three Rivers Press, 1982), 254–55.

12胡克关于行星运动的工作:例如 Michael Nauenberg,《罗伯特·胡克对轨道动力学的开创性贡献》,《物理学展望》7(2005):1-31。Nauenberg 对胡克的工作以及牛顿如何利用它进行了详细的研究。给予胡克应有的赞誉是很重要的,尽管我倾向于认为 Nauenberg 高估了胡克的数学能力,因为他的论证基于一个单一的构造,这个构造是在 1685 年他很可能看过牛顿的初步论文《运动论》之后做出的。无论如何,胡克构造了由与距离成正比的力产生的轨道,虽然他以一种新颖的方式做到了这一点,但与《自然哲学的数学原理》针对各种力和运动的数百次计算相比,这只是一次计算。

12.Hooke’s work on planetary motion: e.g., Michael Nauenberg, “Robert Hooke’s Seminal Contributions to Orbital Dynamics,” Physics in Perspective 7 (2005): 1–31. Nauenberg has made a detailed study of Hooke’s work and how Newton made use of it. It is important to give Hooke the credit he deserves, although I tend to think Nauenberg overplays Hooke’s mathematical ability, given that his argument is based on a single construction, made in 1685 after he had most likely seen Newton’s preliminary paper De Motu. Either way, Hooke constructed the orbit resulting from a force that varies directly with distance, and while he did it in a novel way, it is one calculation compared with the hundreds in Principia, for all kinds of forces and motions.

13牛顿“干计算器”:写给哈雷的信,引自 Nauenberg 的《罗伯特·胡克》,第 7 页。Nauenberg 称这是一篇“长篇大论”,鉴于《自然哲学的数学原理》与胡克的贡献相比范围更广(参见前注),在我看来,他为胡克辩护得有点太激烈了。现代数学,计算与创造力:Patrick Bangert 的异想天开的报告——发表在《澳大利亚数学学会公报》第 32 期第 3 期(2005 年 7 月)——表明许多数学家认为他们的学科与模式、语言、艺术或逻辑的关系比应用更大。同样,2021 年 7 月,Ole Warnaar 在《公报》(第 48 卷第 3 期)上报道了学会成员对澳大利亚数学课程拟议修订的反馈:批评包括其过于功利的方法。有效地运用数学当然是令人兴奋的,而且对社会至关重要,Warnaar 也对教授这种技能表示赞赏;但他也哀叹“人们没有做出足够的努力来传达数学的内在美以及学习和理解新的数学概念所能获得的乐趣。”

13.Newton “dry calculators”: letter to Halley quoted in Nauenberg, “Robert Hooke,” 7. Nauenberg calls this a “diatribe,” which suggests to me, in light of the breadth of Principia compared with Hooke’s contribution (cf. previous note), that he is making his case for Hooke a little too vehemently. Modern mathematics, computation versus creativity: Patrick Bangert’s whimsical report—published in the Australian Mathematical Society’s Gazette 32, no. 3 ( July 2005)—suggests that many mathematicians think their subject has more to do with patterns, language, art, or logic than applications. Similarly, in July 2021, Ole Warnaar reported in the Gazette (vol. 48, no. 3) on feedback from the society’s members regarding proposed revisions to the Australian mathematics curriculum: criticisms included its excessively utilitarian approach. It’s certainly exciting, and socially vital, to apply mathematics usefully, and Warnaar applauds teaching such skills; but he also laments that “not enough effort has been made to try to convey the intrinsic beauty of mathematics and the enjoyment one can derive from learning and understanding new mathematical concepts.”

14在蒸发过程中,热量会增加水分子的动能,这样它们就可以摆脱在液体状态下将分子结合在一起的电键。紫色布料将较冷的紫色光反射到我们的眼睛,吸收其他较暖的颜色。所以它干得最快,因为它比床单上的其他颜色吸收热量更快。杜·夏特莱的研究成果是她 1738 年发表的《论火》Dissertation sur la nature et la propagation du feu)的扩展版,发表于 1744 年(巴黎:Chez Prault Fils)。

14.During evaporation, heat raises the kinetic energy of the water molecules so they can escape the electrical bonds that bound the molecules together as a liquid. The violet cloth reflects cooler violet light to our eyes, absorbing the other warmer colours. So it dries fastest because it absorbs heat more quickly than the other colours on the bedsheet. Du Châtelet’s result, in an expanded version of her 1738 Essay on Fire (Dissertation sur la nature et la propagation du feu), was published in 1744 (Paris: Chez Prault Fils).

15牛顿在《数学原理》中几何版本的微积分:例如,尝试将牛顿在第一卷中对命题 39 的证明翻译成现代符号。他想要找到一个在向心力作用下的下落物体的速度,他将力定义为与速度增量 ( I ) 除以时间增量成正比,这是dv / dt的几何/微分版本;但他也以几乎现代的方式找到了v 2的导数,实际上写成 [( v + I ) 2v 2 ]/Δ y得到 2 vdv / dy。(他使用v 2是因为他有效地证明了一个关于动能是力所做功的定理。)

15.Newton’s geometrical version of calculus in Principia: For example, try translating into modern symbols Newton’s proof of proposition 39 in book 1. He wants to find the velocity of a falling body under a centripetal force, and he defines force as proportional to the increment of velocity (I) divided by the increment of time, a geometric/differential version of dv/dt; but he also finds the derivative of v2 in an almost modern way, writing in effect [(v + I)2v2]/Δy to get 2vdv/dy. (He uses v2 because he’s effectively proving a theorem about kinetic energy as the work done by the force.)

16有关沙特莱 (du Châtelet) 的牛顿理论的更多信息,请参阅我的《逻辑的诱惑:埃米莉·沙特莱 (Émilie du Châtelet)、玛丽·萨默维尔 (Mary Somerville) 和牛顿革命》 (纽约:牛津大学出版社,2012 年)。

16.For more on du Châtelet’s Newtonian work, see my Seduced by Logic: Émilie du Châtelet, Mary Somerville and the Newtonian Revolution (New York: Oxford University Press, 2012).

第三章

CHAPTER 3

1 .SI 单位:Système International d'unités(国际单位制)在国际上缩写为 SI。

1.SI unit: the Système international d’unités—the International System of Units—is abbreviated internationally as SI.

2关于《机械问题》及其影响:David Marshall Miller,《从伪亚里士多德到牛顿的平行四边形规则》,《精确科学史档案》71(2017 年):157-91,特别是 161-66。按照现代标准,《机械问题》仅包含原始平行四边形规则,但即便如此,也很少有人理解其重要性。

2.On Questiones Mechanicae and its influence: David Marshall Miller, “The Parallelogram Rule from Pseudo-Aristotle to Newton,” Archive for History of Exact Sciences 71 (2017): 157–91, esp. 161–66. By modern standards, the Questiones contains only a proto-parallelogram rule, but even then, few grasped its importance.

3 .塔尔塔利亚:他的 45° 计算假设您是从地面射击,而不是从高处射击。“Vituperative” 作品:引自迈克尔布鲁克斯 (Michael Brooks) 的《The Art of More》(墨尔本:Scribe,2021 年),第 94 页。此处和下一段,我要感谢布鲁克斯,还要感谢 JJ O'Connor 和 EF Robertson 在圣安德鲁斯大学 MacTutor 上发表的关于塔尔塔利亚的文章,https://mathshistory.st-andrews.ac.uk/Biographies/Tartaglia/# :~:text=Quick%20Info&text=Tartaglia%20as%20an%20Italian%20mathematician,published%20in%20Cardan%27s%20Ars%20Magna 。

3.Tartaglia: his 45° calculation assumes you’re firing from the ground rather than from a height. “Vituperative” work: quoted in Michael Brooks, The Art of More (Melbourne: Scribe, 2021), 94. Here and in the next paragraph, I’m indebted to Brooks, and also to J. J. O’Connor and E. F. Robertson’s University of St Andrews MacTutor article on Tartaglia, https://mathshistory.st-andrews.ac.uk/Biographies/Tartaglia/#:~:text=Quick%20Info&text=Tartaglia%20as%20an%20Italian%20mathematician,published%20in%20Cardan%27s%20Ars%20Magna.

4 .Miller,“平行四边形规则”,164。

4.Miller, “Parallelogram Rule,” 164.

5约瑟夫·贾瑞特(Joseph Jarrett)在《代数与战争艺术:马洛的《帖木儿一世和二世》中的军事数学》一文中写道,马洛能够通过应用一种类似于哈里奥特开创的代数符号的紧凑表示法,在小舞台上表现巨大的战争场景。贾瑞特还让我注意到了阿尔特多弗的战争画

5.Joseph Jarrett—in “Algebra and the Art of War: Marlowe’s Military Mathematics in Marlowe’s ‘Tamburlaine I and II,’” Cahiers Élizabéthain 95, no. 1 (2018): 19–39—suggests that Marlowe was able to represent huge battle scenes on a small stage by applying a kind of compact representation similar to the algebraic symbolism pioneered by Harriot. Jarrett also alerted me to Altdorfer’s battle paintings.

6 .伽利略和哈里奥特对下落运动和水平投影的理解是正确的,但他们将抛射体的斜运动视为单一的减速运动,而不是多个独立分量:Matthias Schemmel,《托马斯·哈里奥特作为英国的伽利略:早期现代力学中共享知识的力量》,载托马斯·哈里奥特,第二卷,《早期现代英格兰的数学、探索和自然哲学》,罗伯特·福克斯主编(法纳姆,萨里:阿什盖特出版社,2012 年),第 89–111 页,尤其是第 95、97 页。

6.Galileo and Harriot got falling motion and horizontal projection right, but they treated the oblique motion of a projectile as a single decelerated motion rather than in terms of independent components: Matthias Schemmel, “Thomas Harriot as an English Galileo: The Force of Shared Knowledge in Early Modern Mechanics,” in Thomas Harriot, vol. 2, Mathematics, Exploration and Natural Philosophy in Early Modern England, ed. Robert Fox (Farnham, Surrey: Ashgate, 2012), 89–111, esp. 95, 97.

7伽利略、斯蒂文、笛卡尔和平行四边形规则:米勒,《平行四边形规则》,166、167、170、183、186。

7.Galileo, Stevin, and Descartes and the parallelogram rule: Miller, “Parallelogram Rule,” 166, 167, 170, 183, 186.

8Harriot 的计算(图 3.3:在每个图中,左侧的实心圆aA表示两个球的起点。它们在中间相撞,然后反弹到虚线圆所示的位置。图下的计算考虑了球的速度和质量:Harriot 正在观察给定时间间隔x内发生的情况,因此速度以球之间线的方向和长度来表示,类似于我们今天对矢量的处理。

8.Harriot’s calculations (fig. 3.3): In each diagram the solid circles a and A on the left show the starting points of the two balls. They collide in the middle and rebound to the positions shown with dotted circles. The calculations below the diagram take account of the velocities and masses of the balls: Harriot is looking at what happens in a given time interval x, so the velocities are expressed in terms of the directions and lengths of the lines between the balls, analogously to what we would do with vectors today.

哈里奥特仔细解释了他使用平行四边形规则的原因。例如,他说如果没有发生碰撞,那么在第二个时间间隔x内,球a将继续以相同的速度沿同一直线ab移动——因此他凭直觉推断出了第一运动定律。然而,在发生碰撞时,该运动被“平移”到平行线fc——他强调,这不是物理上的,而是为了“合成视在运动”。对于第二个球,运动AB被平移到FC

Harriot explains his use of the parallelogram rule carefully. For instance, he says that if there had been no collision, then in the second time interval x the ball a would have continued moving with the same speed in the same line ab—so he has intuited the first law of motion. On collision, though, this motion is “translated” to the parallel line fc—not physically, he stresses, but for the purposes of “compos[ing] the apparent motion.” Similarly for the second ball, the motion AB is translated to FC.

fF是根据碰撞前后每个球的运动组合得出的。例如,对于第一个球,bf等于(我们称之为)两个矢量之和:(减去)球的初始运动的垂直分量bd(即,它好像只是从一个质量相同的静止球上反弹的运动)加上较大球的动量(我们称之为)赋予的额外运动的垂直分量df (= gb )。

The points f and F are found from the composition of the motion of each ball before and after the collision. For example, for the first ball, bf equals the sum of (what we would call) two vectors: (minus) the vertical component of the ball’s initial motion, bd—that is, the motion as if it had simply rebounded from a stationary ball of equal mass—plus the vertical component of the extra motion imparted by the larger ball’s momentum (as we would put it), df(=gb).

用动量和动能守恒可以更轻松地完成,但哈里奥特没有这些概念。然而,他的图的对称性表明他假设了“动量”守恒,图下方的计算——他的b¯¯代表两个球的速度和它们的质量——与我们在动量守恒计算中的结果基本相同。注意

It can be done more easily using conservation of momentum and kinetic energy, but these concepts weren’t available to Harriot. The symmetry of his diagram, however, shows that he is assuming the conservation of “impetus,” and the calculations below the diagram—where his b¯,B¯ represent the velocities of the two balls and b, B their masses—are essentially the same as in our conservation of momentum calculations. Note that

图像

他的符号是b¯+¯b+/。(他在这个词上犯了一个小错误。)

is his notation for b¯+B¯b+B/B. (He’s made a slight error in this term.)

Harriot 的小错误(加上对其论文的分析):Johannes Lohne,《论 Thomas Harriot》,《精确科学史档案》 20 卷,第 3/4 期(1979 年):189–312,以及 Jon V. Pepper,《Harriot 的撞击理论手稿》,《科学年鉴》 33 卷,第 2 期(1976 年):131–51,DOI:10.1080/00033797600200191。

Harriot’s minor error (plus analysis of his paper): Johannes Lohne, “Essays on Thomas Harriot,” Archive for History of Exact Sciences 20, nos. 3/4 (1979): 189–312, and Jon V. Pepper, “Harriot’s Manuscript on the Theory of Impacts,” Annals of Science 33, no. 2 (1976): 131–51, DOI: 10.1080/00033797600200191.

9“仿佛”是哈里奥特的措辞,正如米勒在“平行四边形规则”第 158 页等现代记述中所用到的一样。哈里奥特的论文最初发表于 20 世纪 70 年代:请参阅 Lohne 的“托马斯·哈里奥特论文集”,其中有从拉丁文原文翻译成英文的内容。

9.“As if ” is Harriot’s wording, just as in modern accounts such as Miller, “Parallelogram Rule,” 158. Harriot’s paper was first published in the 1970s: see Lohne, “Essays on Thomas Harriot,” for an English translation from the original Latin.

10Wallis 和 Harriot:Jacqueline Stedall,《光荣被剥夺:Thomas Harriot 及其代数的死后不幸》,《精确科学史档案》 54,第 6 期(2000 年 6 月):483。Wallis和 Fermat:Miller,《平行四边形规则》,171–72,186–87。

10.Wallis and Harriot: Jacqueline Stedall, “Rob’d of Glories: The Posthumous Misfortunes of Thomas Harriot and His Algebra,” Archive for History of Exact Sciences 54, no. 6 ( June 2000): 483. Wallis and Fermat: Miller, “Parallelogram Rule,” 171–72, 186–87.

11沃利斯试图找到一种方法来思考x 2 + 2 bx + c 2 = 0形式的二次方程的复数解——如果你还记得可以通过完成平方推导出的二次公式,你就会知道=b±b2c2bc时,沃利斯找到了一种方法将两个解表示为实数线上的点——因此,他对b < c时得到的复数解也做了类似的尝试。他通过考虑以下公式来避免明确处理i=b±c2b2,他用繁琐的三角形结构来表示他的解,而不是像我们今天那样将它们表示为复平面上的点。约翰·斯蒂尔威尔在《数学及其历史》(纽约:Springer-Verlag,1989 年)图 13.3 中展示了沃利斯的尝试及其缺陷。

11.Wallis was trying to find a way to think about complex solutions to quadratic equations of the form x2 + 2bx + c2 = 0—and if you remember the quadratic formula that you can deduce by completing the square, you’ll know that x=b±b2c2. When bc, Wallis found a way to represent the two solutions as points on the real number line—so he tried something similar for the complex solutions you get when b < c. He avoided having to deal explicitly with i by considering x=b±c2b2, and he’d used cumbersome constructions with triangles to represent his solutions rather than representing them as points on a complex plane as we would do today. John Stillwell shows Wallis’s attempt and its flaws in fig. 13.3, Mathematics and Its History (New York: Springer-Verlag, 1989).

12欧拉对e的定义的现代版本是1+1nn。它的概念最早出现在雅各布·伯努利(Johann/Jean 的哥哥)对复利的研究中,甚至更早,尽管隐含地,出现在约翰·纳皮尔的对数和托马斯·哈里奥特未发表的连续复利和子午线部分的计算中。但正是欧拉认识到了这个数字的力量并引起了人们的适当关注。他还把它写成了泰勒级数,这是计算e的十进制近似值的方式,例如您的计算器上的那个(我的计算器给出 2.718281828)。

12.The modern version of Euler’s definition of e is limx1+1nn. The idea of it first arose in a study of compound interest by Jakob (or Jacques) Bernoulli ( Johann/Jean’s older brother), and even earlier, although implicitly, in John Napier’s logarithms and Thomas Harriot’s unpublished calculations for continuously compounding interest and meridional parts. But it was Euler who recognised the power of this number and brought it to proper attention. He also wrote it as a Taylor series, which is the way decimal approximations of e are worked out, such as the one on your calculator (mine gives 2.718281828).

13关于欧拉及其身份:Ed Sandifer,《欧拉是如何做到的》,MAA Online2007 年 8 月。另请参阅 Carl Boyer 的《数学史》,Uta Merzbach 修订版(纽约:John Wiley and Sons,1991 年),第 443–44 页(以及关于欧拉和大学数学的第 441–42 页)。De Moivre 和 Newton:Orlando Merlino,《复数简史》,罗德岛大学,2006 年 1 月。

13.On Euler and his identity: Ed Sandifer, “How Euler Did It,” MAA Online, August 2007. See also Carl Boyer, History of Mathematics, rev. Uta Merzbach (New York: John Wiley and Sons, 1991), 443–44 (and 441–42 on Euler and uni maths). De Moivre and Newton: Orlando Merlino, “A Short History of Complex Numbers,” University of Rhode Island, January 2006.

14欧拉和费马最后定理:后来的数学家发现,费马没有证明他的证明中的一个步骤,但是该步骤本身是正确的;这一点以及欧拉在证明中使用复数的原因在 Harold M. Edwards 的《费马最后定理》中得到了很好的解释,《科学美国人》 239,第 4 期(1978 年 10 月):104-23。

14.Euler and Fermat’s last theorem: Later mathematicians found that Fermat hadn’t proven one of the steps in his proof, but the step itself was correct; this, and Euler’s use of complex numbers in the proof, is explained well in Harold M. Edwards, “Fermat’s Last Theorem,” Scientific American 239, no. 4 (October 1978): 104–23.

15欧拉/达朗贝尔:史迪威,《数学及其历史》,202。索菲·热尔曼并没有发表她关于费马大定理的著作,但勒让德将他在n = 5 时的证明中所用的一个结果归功于她。

15.Euler/d’Alembert: Stillwell, Mathematics and Its History, 202. Sophie Germain didn’t publish her work on Fermat’s theorem, but Legendre credited her with a result he used in his proof when n = 5.

16汉密尔顿的苹果和橘子:凯伦·亨格·帕歇尔,《抽象代数的发展》,《普林斯顿数学指南》 ,蒂莫西·高尔斯等主编(新泽西州普林斯顿:普林斯顿大学出版社,2010 年),第 95–106 页,尤其是第 105 页。

16.Hamilton’s apples and oranges: Karen Hunger Parshall, “The Development of Abstract Algebra,” in The Princeton Companion to Mathematics, ed. Timothy Gowers et al. (Princeton, NJ: Princeton University Press, 2010), 95–106, esp. 105.

17高斯引自 Christian Gérini,《阿尔冈德虚数的几何表示》,土伦大学,2009 年 1 月,英文译本。Helen Tomlinson,2017 年 4 月,在线网址:http://www.bibnum.education.fr/sites/default/files/21-argand-analysis.pdf。阿尔冈德的“有向线”亦可参见此处。德·摩根引自 Raymond Flood 和 Robin Wilson,《伟大的数学家》 (伦敦:Arcturus,2011 年),第 143 页。有关德·摩根的更多背景信息,请参阅:Morris Kline,《数学:确定性的丧失》(纽约:牛津大学出版社,1982 年),第 155-56 页。

17.Gauss quoted by Christian Gérini, “Argand’s Geometric Representation of Imaginary Numbers,” University of Toulon, January 2009, English trans. Helen Tomlinson, April 2017, online at http://www.bibnum.education.fr/sites/default/files/21-argand-analysis.pdf. See this also for Argand’s “directed lines.” De Morgan quoted in Raymond Flood and Robin Wilson, The Great Mathematicians (London: Arcturus, 2011), 143. For more context on De Morgan: Morris Kline, Mathematics: The Loss of Certainty (New York: Oxford University Press 1982), 155–56.

18巴贝奇引用自 Dirk Struik 所著《数学简史》(纽约:多佛,1967 年),第 168 页。

18.Babbage quoted in Dirk Struik, A Concise History of Mathematics (New York: Dover, 1967), 168.

19汉密尔顿引用自珍妮特·福利纳 (Janet Folina) 的《牛顿和汉密尔顿:为代数中的真理辩护》,《南方哲学杂志》 50,第 3 期(2012 年):515。

19.Hamilton quoted in Janet Folina, “Newton and Hamilton: In Defense of Truth in Algebra,” Southern Journal of Philosophy 50, no. 3 (2012): 515.

20汉密尔顿从负数/时间科学到复数对再到四元数的过程,在 Teun Koetsier 的《数学史学解释:汉密尔顿四元数案例》一文中进行了详细解释,《科学史与科学哲学研究》第 A 部分第26 卷,第 4 期(1995 年):593-616 页。汉密尔顿的“步骤”作为向量:参见他的《四元数讲座》 (都柏林:霍奇斯和史密斯出版社;伦敦:惠特克出版社;剑桥:麦克米伦出版社,1853 年),第 3 页及后续页。

20.Hamilton’s process, from negative numbers/science of time to complex couples to quaternions, is explained in detail by Teun Koetsier, “Explanation in the Historiography of Mathematics: The Case of Hamilton’s Quaternions,” Studies in History and Philosophy of Science Part A 26, no. 4 (1995): 593–616. Hamilton’s “steps” as vectors: see his Lectures on Quaternions (Dublin: Hodges and Smith; London: Whittaker; and Cambridge: Macmillan, 1853), 3ff.

哈密​​顿在复数中的重要性:帕歇尔,《抽象代数的发展》,105;博耶,《数学史》,583。哈密顿在时间代数中引用牛顿的话:福利纳,《牛顿和哈密顿》,513。

Hamilton’s importance re: complex numbers: Parshall, “Development of Abstract Algebra,” 105; Boyer, History of Mathematics, 583. Hamilton citing Newton on algebra of time: Folina, “Newton and Hamilton,” 513.

21德·摩根 (De Morgan) 论汉密尔顿的情侣关系:引自戴安娜·威尔门特 (Diana Willment),《1600 年至 1840 年的复数》(硕士论文,密德萨斯大学,1985 年),第 102 页。汉密尔顿,象征主义,德·摩根:科特希尔,《解释》,第 610 页。

21.De Morgan on Hamilton’s couples: quoted in Diana Willment, “Complex Numbers from 1600 to 1840” (master’s thesis, Middlesex University, 1985), 102. Hamilton, symbolism, De Morgan: Koetsier, “Explanation,” 610.

22包括华兹华斯在内的文学好友都很欣赏汉密尔顿:迈克尔·J·克罗,《矢量分析史》(印第安纳州圣母大学:圣母大学出版社,1967 年),第 22 页;丹尼尔·布朗,《维多利亚时代科学家的诗歌:风格、科学与废话》(剑桥:剑桥大学出版社,2013 年),第 1 页。玛丽亚·埃奇沃思论汉密尔顿:“埃奇沃思小姐的建议”,爱尔兰皇家学院 (RIA) (博客),2018 年 6 月 24 日,https://www.ria.ie/news/library-library-blog/miss-edgeworth-advises

22.Literary friends, including Wordsworth, admiring Hamilton: Michael J. Crowe, A History of Vector Analysis (Notre Dame, IN: University of Notre Dame Press, 1967), 22; and Daniel Brown, The Poetry of Victorian Scientists: Style, Science and Nonsense (Cambridge: Cambridge University Press, 2013), 1. Maria Edgeworth on Hamilton: “Miss Edgeworth Advises,” Royal Irish Academy (RIA) (blog), June 24, 2018, https://www.ria.ie/news/library-library-blog/miss-edgeworth-advises.

23玛丽亚·埃奇沃思 (Maria Edgeworth)、皮科克 (Peacock) 和萨默维尔 (Somerville ):参见我的《逻辑的诱惑:埃米莉·杜·夏特莱 (Émilie du Châtelet)、玛丽·萨默维尔和牛顿革命》 (纽约:牛津大学出版社,2012 年),第 195 页、第 197–98 页。

23.Maria Edgeworth, Peacock, and Somerville: see my Seduced by Logic: Émilie du Châtelet, Mary Somerville and the Newtonian Revolution (New York: Oxford University Press, 2012), 195, 197–98.

24玛丽亚·埃奇沃思和爱尔兰皇家学院:RIA(博客),“埃奇沃思小姐的建议”和克莱尔·奥哈洛伦,“'没有女士更好':爱尔兰皇家学院和女性成员的接纳”, 18-19世纪社会视角》(爱尔兰历史)第 19 卷,第 6 期(2011 年 11 月/12 月):42–46

24.Maria Edgeworth and Royal Irish Academy: RIA (blog), “Miss Edgeworth Advises,” and Clare O’Halloran, “‘Better without the Ladies’: The Royal Irish Academy and the Admission of Women Members,” 18th–19th Century Social Perspectives (History Ireland) 19, no. 6 (November/December 2011): 42–46.

25汉密尔顿关于向量作为有向线的论述:例如,他的《四元数讲座》,35。

25.Hamilton on vectors as directed lines: e.g., his Lectures on Quaternions, 35.

第四章

CHAPTER 4

1 .爸爸,你会乘三胞胎吗?汉密尔顿写给阿奇博尔德的信,1865 年,载于罗伯特·P·格雷夫斯的《威廉·罗恩·汉密尔顿爵士传》,全 3 卷。(都柏林:霍奇斯,菲吉斯,1882、1885、1889 年),2:434-35,被广泛引用,例如,载于迈克尔·J·克罗的《矢量分析史》(印第安纳州圣母大学:圣母大学出版社,1967 年),29。向“我的孩子们”解释:写给德·摩根的信,1852 年,格雷夫斯的《汉密尔顿传》(1889 年),3:#59,307-8。

1.Papa, can you multiply triplets? Hamilton’s letter to Archibald, 1865, in Robert P. Graves, Life of Sir William Rowan Hamilton, 3 vols. (Dublin: Hodges, Figgis, 1882, 1885, 1889), 2:434–35, widely quoted, e.g., in Michael J. Crowe, A History of Vector Analysis (Notre Dame, IN: University of Notre Dame Press, 1967), 29. Explaining to “my boys”: letter to De Morgan, 1852, Graves, Life of Hamilton (1889), 3: #59, 307–8.

2代数/算术定律:例如,(3 + 2) + 5 = 5 + 5 = 10;但在这种情况下,括号放在哪里并不重要,因为 3 + (2 + 5) = 3 + 7 也等于 10。这称为加法的结合律,乘法也有类似的结合律。类似地,2 × 3 = 3 × 2,2 + 3 = 3 + 2;这是乘法和加法的著名交换律。Peacock 还引入了分配律,a ( b + c ) = ab + bc

2.Laws of algebra/arithmetic: For example, (3 + 2) + 5 = 5 + 5 = 10; but in this case, it doesn’t matter where the brackets go, because 3 + (2 + 5) = 3 + 7 which also equals 10. This is called the associative law for addition, and there’s a similar one for multiplication. Similarly, 2 × 3 = 3 × 2, and 2 + 3 = 3 + 2; this is the famous commutative law for multiplication and addition. Peacock also introduced the distributive law, a(b + c) = ab + bc.

3 .模数定律:对于一个普通复数z = x + iy,汉密尔顿已经证明,你可以通过取平方根来定义这个数的“模数”(大小或“绝对值”)。

3.Law of moduli: With an ordinary complex number z = x + iy, Hamilton had shown that you can define the “modulus” (the magnitude or “absolute value”) of this number by taking the square root of

x + iy)(xiy)= x 2 + y 2

(x + iy)(xiy) = x2 + y2,

其中左边的第二个因子是第一个因子的“共轭”。(这可能早在欧拉时代就已经知道了。)结果还表明两个复数乘积的模等于两个模的乘积——汉密尔顿称之为“模数定律”,在今天的教科书中用符号表示为 | zw | = | z || w |。

where the second factor on the left-hand side is the “conjugate” of the first. (This may have been known as early as Euler.) It also turns out that the modulus of the product of two complex numbers equals the product of the two moduli—what Hamilton referred to as “the law of moduli,” written symbolically in today’s textbooks as |zw| = |z||w|.

例如,设z = x + iy, w = a + ib。则

For example, let z = x + iy, w = a + ib. Then

=|+一个+b|=|一个-b+b+一个|=一个b2+b+一个2

zw=|(x+iy)(a+ib)|=|(xa-yb)+i(xb+ya)|=(xayb)2+(xb+ya)2

and

=2+2一个2+b2=一个b2+b+一个2

zw=x2+y2a2+b2=xayb2+xb+ya2.

因此 | zw | = | z || w |,模量定律在二维中成立。

So |zw| = |z||w|, and the law of moduli holds in two dimensions.

但是,如果你尝试用x + iy + jza + ib + jc来做同样的操作,为了使模数定律成立,你必须对x、y、a、b之间的关系做出简化假设(汉密尔顿也尝试过,但这破坏了模数定律的普遍性),或者对i、j、ijji之间的关系做出简化假设。请参阅叙述了解汉密尔顿接下来做了什么。有关汉密尔顿的流程,包括写给格雷夫斯的信,请参阅 Teun Koetsier 的《数学史学解释:汉密尔顿四元数案例》,《科学史与哲学研究》第 A 卷第26 卷,第 4 期(1995 年):593–616 页。

If you try this with x + iy + jz and a + ib + jc, however, for the law of moduli to hold you have to make simplifying assumptions about the relationships between x, y, a, b—which Hamilton tried, but which destroys the generality of the law of moduli—or about the relationships between i, j, ij, and ji. See the narrative for what Hamilton did next. And see Teun Koetsier, “Explanation in the Historiography of Mathematics: The Case of Hamilton’s Quaternions,” Studies in the History and Philosophy of Science Part A 26, no. 4 (1995): 593–616, for Hamilton’s process, including letters to Graves.

4 .汉密尔顿在他的《四元数讲座》序言中解释了这一过程。

4.Hamilton explained this process in the preface to his Lectures on Quaternions.

5奥古斯都·德·摩根,《牛顿生平与工作论文集》,由菲利普·乔丹编辑,附有注释和附录(芝加哥:Open Court,1914 年)。德·摩根的生平注释:莱斯利·斯蒂芬,《国家传记词典》第 14 卷(1885-1900 年),德·摩根 sv;顺便说一句,斯蒂芬是著名小说家弗吉尼亚·伍尔夫的父亲。另见卡尔·博耶,《数学史》,由乌塔·默茨巴赫修订(纽约:John Wiley and Sons,1991 年),第 581 页。

5.Augustus De Morgan, Essays on the Life and Work of Newton, edited, with notes and appendices, by Philip Jourdain (Chicago: Open Court, 1914). Biographical notes on De Morgan: Leslie Stephen, Dictionary of National Biography 14 (1885–1900), s.v. De Morgan; incidentally, Stephen was the father of the famous novelist Virginia Woolf. Also see, e.g., Carl Boyer, History of Mathematics, rev. Uta Merzbach (New York: John Wiley and Sons, 1991), 581.

6 .“深深崇敬”:亚历山大·麦克法兰,《十位英国数学家讲座》(伦敦:查普曼和霍尔,1916 年),第 3 章(摘自 1901 年的讲座)。

6.“Deeply reverential”: Alexander MacFarlane, Lectures on Ten British Mathematicians (London: Chapman and Hall, 1916), chap. 3 (from a lecture delivered in 1901).

7最近的学术研究追溯了有关汉密尔顿的流言蜚语的演变,并将其置于女性角色和社会约束的背景下,描绘了他和海伦生活的更正面的画面:安妮·范·韦尔登和斯蒂芬·韦普斯特,《关于天才的最八卦:威廉·罗恩·汉密尔顿爵士》,BSHM Bulletin 33,第 1 期(2018 年):2-20。他们还给出了比早期记录更平衡的描述,例如,将汉密尔顿醉酒的情节放在禁酒运动的背景下。

7.Recent scholarship has traced the evolution of the gossip about Hamilton, and put it in the context of women’s roles and social constraints and painting a more positive picture of his and Helen’s life: Anne van Weerden and Stephen Wepster, “A Most Gossiped about Genius: Sir William Rowan Hamilton,” BSHM Bulletin 33, no. 1 (2018): 2–20. They also give a more balanced account than earlier records suggested, e.g., putting an episode where Hamilton was supposed to be violently drunk into the context of the temperance movement.

8德·摩根和洛夫莱斯:克里斯托弗·霍林斯、乌尔苏拉·马丁和阿德里安·赖斯的两篇论文:《艾达·洛夫莱斯的早期数学教育》,《BHSM 通讯》第 32 期第 3 期(2017 年):第 221-34 页,以及《洛夫莱斯-德·摩根通信:批判性重新评价》,《数学史》第 44 期(2017 年):第 202-31 页。请注意,巴贝奇的“差分机”超前于时代,从未投入生产。

8.De Morgan and Lovelace: two papers by Christopher Hollings, Ursula Martin, and Adrian Rice: “The Early Mathematical Education of Ada Lovelace,” BHSM Bulletin 32, no. 3 (2017): 221–34, and “Lovelace-De Morgan Correspondence: A Critical Re-appraisal,” Historia Mathematica 44 (2017): 202–31. Note that Babbage’s “difference engine” was ahead of its time and never went into production.

9德·摩根在珍妮特·福利纳的《牛顿和汉密尔顿:为代数中的真理辩护》中引用了这句话,《南方哲学杂志》 50,第 3 期(2012 年):511。请注意,汉密尔顿和德·摩根对代数基础有不同的看法(511-12)。

9.De Morgan quoted in Janet Folina, “Newton and Hamilton: In Defense of Truth in Algebra,” Southern Journal of Philosophy 50, no. 3 (2012): 511. Note that Hamilton and De Morgan had different approaches to algebraic foundations (511–12).

10汉密尔顿致德摩根,1841 年,引用自迈克尔·J·克罗所著《矢量分析史》(印第安纳州圣母大学:圣母大学出版社,1967 年),第 27 页。

10.Hamilton to De Morgan, 1841, quoted, e.g., Michael J. Crowe, A History of Vector Analysis (Notre Dame, IN: University of Notre Dame Press, 1967), 27.

11调用 k = ij:对于三元组a + ib + jcx + iy + jz,模数定律表明

11.Invoking k = ij: For triples a + ib + jc and x + iy + jz, the law of moduli says that

|(a + ib + jc)(x + iy + jz)| = | a + ib + jc || x + iy + jz |。

|(a + ib + jc)(x + iy + jz)| = |a + ib + jc||x + iy + jz|.

右侧(RHS)就是

The right-hand side (RHS) is just

a2 + b2 + c2 x2 + y2 + z2

(a2 + b2 + c2)(x2 + y2 + z2).

现在对于左侧(LHS):假设ij = − ji,并在展开括号后考虑 LHS:

Now for the left-hand side (LHS): assume ij = −ji, and consider the LHS when you’ve expanded the brackets:

| axbycz + i ( ay + bx ) + j ( az + cx ) + ij ( bzcy )| =

( axbycz ) 2 + ( ay + bx ) 2 + ( az + cx ) 2 + ( bzcy ) 2

|axbycz + i(ay + bx) + j(az + cx) + ij(bzcy)| =

(axbycz)2 + (ay + bx)2 + (az + cx)2 + (bzcy)2.

但是,你只能通过将ij ( bzcy ) 的共轭纳入模数定义中来获得最后一个项(需要与 RHS 平衡) ,就好像 ij 是像 i 和 j 一样的复数。这导致 Hamilton 假设,为了解决三重乘法问题,他必须调用第三个虚向量k = ij

But you only get this last term—which you need to balance with the RHS—by incorporating the conjugate of ij(bzcy) into the modulus definition, as if ij were a complex number like i and j. This is what led Hamilton to suppose that to solve his problems of triplet multiplication, he had to invoke a third imaginary vector k = ij.

汉密尔顿的“电路”:我稍微修改了“闭合”的时态;参见汉密尔顿 1865 年写给阿奇博尔德的信,引自克罗的《矢量史》,第 29 页。汉密尔顿写给格雷夫斯的信:引自 BL van der Waerden 的《汉密尔顿对四元数的发现》,《数学杂志》第 49 卷,第 5 期(1976 年 11 月):第 227-34 页,特别是第 230 页。

Hamilton’s “electric circuit”: I’ve slightly modified the tense on “closed”; cf. Hamilton’s 1865 letter to Archibald, quoted in Crowe, History of Vectors, 29. Hamilton to Graves: quoted in B. L. van der Waerden, “Hamilton’s Discovery of Quaternions,” Mathematics Magazine 49, no. 5 (November 1976): 227–34, esp. 230.

12德·摩根引用自 Folina,《牛顿和汉密尔顿》,第 505 页。华兹华斯对汉密尔顿平庸诗歌的看法:丹尼尔·布朗,《维多利亚科学家的诗歌:风格、科学和废话》(剑桥:剑桥大学出版社,2013 年),第 1-2 页。薛定谔:克罗,《矢量分析史》,第 17 页。薛定谔指的是现在所谓的哈密顿动力学,这是运动定律的另一种、无坐标形式,相当于牛顿方法但更加灵活。

12.De Morgan quoted in Folina, “Newton and Hamilton,” 505. Wordsworth on Hamilton’s mediocre poetry: Daniel Brown, The Poetry of Victorian Scientists: Style, Science and Nonsense (Cambridge: Cambridge University Press, 2013), 1–2. Schrödinger: Crowe, History of Vector Analysis, 17. Schrödinger was referring to what is now called Hamiltonian dynamics, an alternative, coordinate-free form of the laws of motion, equivalent to Newton’s approach but more flexible.

13只有当你假设只能在字符串开头或结尾处取消项,并且对的乘积是反对交换的,玩弄汉密尔顿的涂鸦才有效。因此,要从j 2 = ijk找到j,请重写为j 2 = − jik,然后从两边取消j ,你会得到j = − ik = ki

13.Playing with Hamilton’s graffiti only works if you assume that you can cancel terms only at the beginning or end of the string, and if products of pairs are anticommutative. So, to find j from j2 = ijk, rewrite as j2 = −jik, and then, canceling j from both sides, you have j = −ik = ki.

14标量积只是将分量相乘,因此pq = p 1 q 1 + p 2 q 2 + p 3 q 3是一个数字或标量,而不是矢量。(在 Hamilton 的完整四元数积中, pq前面有一个减号,这将引起争议,我们将在第 7 章中看到。)

14.Scalar products just multiply the components, so that pq = p1q1 + p2q2 + p3q3 is a number, or scalar, not a vector. (In Hamilton’s full quaternion product, there’s a minus sign in front of pq, which will prove controversial, as we’ll see in chap. 7.)

向量积得到向量; p × q的分量形式最容易记住,并且可以通过行列式计算123123

Vector products give vectors; the component form of p × q is easiest to remember and calculate from the determinant ijkp1p2p3q1q2q3

15凯莱和四元数:克罗,《向量分析史》,第 35 页。凯莱的生平:托尼·克里利,《阿瑟·凯莱:未选择的路》,《数学情报》第 20 卷,第 4 期(1998 年):第 49–53 页;克里利还撰写了《大英百科全书》中有关凯莱的条目。

15.Cayley and quaternions: Crowe, History of Vector Analysis, 35. Cayley’s life: Tony Crilly, “Arthur Cayley: The Road Not Taken,” Mathematical Intelligencer 20, no. 4 (1998): 49–53; Crilly also wrote the Britannica entry on Cayley.

16例如,在 Erwin Kreyszig 的《高级工程数学》(纽约:Wiley,1993 年),976 页中给出了高斯消元法的计算机算法。

16.A computer algorithm for Gaussian elimination is given, e.g., in Erwin Kreyszig, Advanced Engineering Mathematics (New York: Wiley, 1993), 976.

17西尔维斯特“不变量”:克里利,《阿瑟·凯莱:未选择的路》,51。

17.Sylvester “invariants”: Crilly, “Arthur Cayley: The Road Not Taken,” 51.

18布尔和凯莱:Tony Crilly,《凯莱不变理论的兴起》,《数学史》 13(1986):241-54。

18.Boole and Cayley: Tony Crilly, “The Rise of Cayley’s Invariant Theory,” Historia Mathematica 13 (1986): 241–54.

19尤妮丝·富特,《影响太阳光线热量的环境》,《美国科学与艺术杂志》(1856 年):第 382-83 页。约翰·廷德尔(John Tyndall)似乎在几年后就独立地将物理学融入了富特的经验发现中:参见罗兰·杰克逊(Roland Jackson),《约翰·廷德尔:被遗忘的气候科学共同创始人》,《对话》,2020 年 7 月 31 日。让-巴蒂斯特·傅立叶(Jean-Baptiste Fourier)于 1820 年首次研究了大气的加热效应,但他是在计算地球温度的背景下进行的。

19.Eunice Foote, “Circumstances Affecting the Heat of the Sun’s Rays,” American Journal of Science and Arts (1856): 382–83. John Tyndall, apparently independently, put the physics into Foote’s empirical discovery just a few years later: see Roland Jackson, “John Tyndall: The Forgotten Cofounder of Climate Science,” The Conversation, July 31, 2020. Jean-Baptiste Fourier was the first to look at the heating effect of the atmosphere, in 1820, but he did it in the context of calculating Earth’s temperature.

20搜索引擎:推导两个向量 a 和 b 之间的角度所需的标量积形式为ab = | a || b |cosθ。我的描述参考了 Amy Langville 的精彩介绍“搜索引擎背后的线性代数:重点关注向量空间模型”,《Convergence》,美国数学协会(2006 年 12 月);https://www.maa.org/press/periodicals/loci/joma/the-linear-algebra-behind-search-engines-focus-on-the-vector-space-model

20.Search engines: The form of the scalar product needed to deduce the angle between two vectors a and b is ab = |a||b|cosθ. For my description I’ve drawn on Amy Langville’s excellent introduction, “The Linear Algebra behind Search Engines: Focus on the Vector Space Model,” Convergence, Mathematical Association of America (December 2006); https://www.maa.org/press/periodicals/loci/joma/the-linear-algebra-behind-search-engines-focus-on-the-vector-space-model.

21Google PageRank 算法:我借鉴了康奈尔大学的信息讲座,http://pi.math.cornell.edu/~mec/Winter2009/RalucaRemus/Lecture3/lecture3.html

21.Google PageRank algorithm: I’ve drawn on Cornell University’s informative lecture, http://pi.math.cornell.edu/~mec/Winter2009/RalucaRemus/Lecture3/lecture3.html.

22对人工智能、社交媒体和搜索算法的批判:例如,参见 Cathy O'Neill 的《数学毁灭性武器:大数据如何加剧不平等并威胁民主》(纽约:皇冠出版公司,2017 年);Safiyah Umoja Noble 的《压迫算法》(纽约:纽约大学出版社,2018 年);Shoshana Zuboff 的《监控资本主义时代:在权力新前沿争取人类未来》(伦敦:Profile Books,2019 年);等等。

22.Critiques of AI, social media, and search algorithms: See, e.g., Cathy O’Neill, Weapons of Math Destruction: How Big Data Increases Inequality and Threatens Democracy (New York: Crown Publishing, 2017); Safiyah Umoja Noble, Algorithms of Oppression (New York: NYU Press, 2018); Shoshana Zuboff, The Age of Surveillance Capitalism: The Fight for a Human Future at the New Frontier of Power (London: Profile Books, 2019); and many others.

23莎拉·弗兰纳里的算法不符合安全协议(参见她的书《代码之道》 [伦敦:Profile Books,2001]),但研究人员仍然相信非交换乘法将被证明是一种有价值的加密工具。

23.Sarah Flannery’s algorithm failed the safety protocol (cf. her book In Code [London: Profile Books, 2001]), but researchers still believe noncommutative multiplication will prove a valuable cryptographic tool.

24四元数旋转:汉密尔顿在他的《四元数讲座》第 269 页(第 282 条)中简要概述了以下方法:

24.Quaternion rotations: Hamilton briefly outlined the following approach in his Lectures on Quaternions, 269 (art. 282):

举一个简单的例子,要绕 i 轴旋转矢量p 可以选择单位四元数U = cos θ + i sin θ,类似于图 4.3中的复数。(这是四元数U = (cosθ, sin θ, 0, 0),因为在这种情况下,旋转轴的jk(或yz )分量为零。)然后,利用欧拉定理,可以得到U = e i θ 。这种形式使乘法更容易——指标定律将乘法变成加法——并清楚地显示了它们与旋转的关系,正如我们在图 3.8中看到的那样。

To take a simple example, to rotate a vector p about the i-axis, you could choose the unit quaternion U = cos θ + i sin θ, by analogy with the complex numbers in my fig. 4.3. (This is the quaternion U = (cosθ, sin θ, 0, 0), because the j and k (or y and z) components of the axis of rotation in this case are zero.) Then, using Euler’s theorem, you have U = eiθ. This form makes multiplications easier—the index laws turn multiplications into additions—and shows clearly how they are related to rotations, as we saw also in figure 3.8.

现在形成一个新的向量,

Now form a new vector,

a = U p U −1 = e i θ p e i θ = e i θ ( ix + jy + kz ) e i θ

a = UpU−1 = eiθp eiθ = eiθ(ix + jy + kz)eiθ.

哈密​​顿一定花了一些时间试验才想出这个组合,因为在阿尔冈平面时不需要乘以U −1(参见图 4.3、3.8)。 (顺便说一下,对于单位四元数,逆元数是复共轭,所以我可以写成a = U p U *而不是a = U p U −1。)从几何上讲,这个U −1因子是需要的,以抵消由于发生在 4 维超空间中而发生的多余旋转,但这超出了我的范围。从代数上讲,我们讨论的是矩阵相似变换的四元数类似物。但您不需要这些技术细节来执行由这种“机器”产生的简单代数运算:

It must have taken Hamilton a bit of experimenting to come up with this combination, for you don’t need to also multiply by U−1 when you’re in the Argand plane (cf. figs. 4.3, 3.8). (By the way, for unit quaternions, the inverse is the complex conjugate, so I could have written a = UpU* rather than a = UpU−1.) Geometrically, this U−1 factor is needed to counteract an extraneous rotation that happens because it’s taking place in a 4-D hyperspace, but that is beyond my scope here. Algebraically we’re talking about the quaternion analogue of a matrix similarity transformation. But you don’t need these technicalities to carry out the simple algebra that results from this “machinery”:

如果用汉密尔顿定义ij代替k,则得到

If you replace k by Hamilton’s definition ij, you get

a = e i θ ( ix + jy + kz ) e i θ

= e i θ ( ix + ( y + iz ) j ) e i θ

= e i θ ( ix ) e i θ + e i θ ( y + iz ) je i θ

a = eiθ(ix + jy + kz)eiθ

= eiθ(ix + (y + iz)j)eiθ

= eiθ(ix)eiθ + eiθ(y + iz)jeiθ.

现在到了最有趣的部分:记住e i θ = cos θ − i sin θ,最后一项中的je i θ变成

Now comes the nifty part: remembering that eiθ = cos θ − i sin θ, the jeiθ in the last term becomes

j (cos θ − i sin θ) = j cos θ − ji sin θ。

j(cos θ − i sin θ) = j cos θ − ji sin θ.

但汉密尔顿定义ij = k = − ji,所以我们有

But Hamilton defined ij = k = −ji, so we have

j cos θ − ji sin θ = j cos θ + ij sin θ = (cos θ) j + ( i sin θ) j = (cos θ + i sin θ) j = e i θ j

j cos θ − ji sin θ = j cos θ + ij sin θ = (cos θ)j + (i sin θ)j = (cos θ + i sin θ)j = eiθj.

(我写成j cos θ = (cos θ) j是因为 cosθ 只是一个实数或标量;只有当两个数都是复数时,乘法才不一定是交换的——就像当ij = k = − ji时一样。对于j sin θ = (sin θ) j也是如此。)

(I wrote j cos θ = (cos θ)j because cosθ is just a real number or scalar; it is only when both numbers are complex that multiplications are not necessarily commutative—as when ij = k = −ji. Similarly for j sin θ = (sin θ)j.)

最后,利用指标定律和哈密顿规则对i, j, k的乘积进行计算,我们得到了向量p的旋转版本:

Finally, using index laws and Hamilton’s rules for products of the i, j, k, we have the rotated version of the vector p:

a = xi + e i ( y + zi ) j = xi + e i ( yj + zk )。

a = xi + ei( y + zi)j = xi + ei(yj + zk).

这看起来是对的,因为p已经绕i轴旋转,所以它的i分量没有变化,因为它只在j - k平面上移动。e i 因子表明pjk分量确实在j - k平面上旋转了 2θ 角。(因此,如果要旋转 θ,请选择单位四元数为U = e i θ/2。)

This looks right, because p has been rotated about the i-axis, so its i-component doesn’t change, for it moves only in the j-k plane. And the ei factor shows that p’s j and k components have, indeed, been rotated in the j-k plane, by an angle of 2θ. (So if you want to rotate by θ, choose the unit quaternion to be U = eiθ/2.)

要绕单位向量u = ai + bj + ck方向的任意轴旋转,而不是像上述计算中那样绕i旋转,请类比欧拉公式,将U = cos θ + u sin θ = e u θ。(您可以通过比较等式两边表达式的级数来证明欧拉公式。)我上面的说明是汉密尔顿大纲的扩展版本,也是富勒顿学院约翰·韦尔塔的讲义“四元数简介”中的示例。

To rotate about an arbitrary axis in the direction of a unit vector u = ai + bj + ck, rather than just i as in the above calculation, put U = cos θ + u sin θ = euθ, by analogy with Euler’s formula. (You can prove Euler’s formula by comparing the series for the expressions on each side of the equation.) My account in the above is an expanded version of Hamilton’s outline, and of the example in the lecture notes, “Introducing the Quaternions,” by John Huerta, Fullerton College.

或者,你也可以通过使用四元数乘法得到的标量和矢量积来展开a = U p U −1来进行上述计算

Alternatively, you can do the above calculations by expanding a = UpU−1 using the scalar and vector products that come from quaternion multiplication, as I showed in the narrative:

PQ = wapq + wq + ap + p × q

PQ = wapq + wq + ap + p × q.

25有关矩阵产生万向锁的示例,请参阅 Justin Wyss-Gallivent 的 MATH431 讲座“万向锁”,2021 年 11 月 3 日:http://www.math.umd.edu/~immortal/MATH431/book/ch_gimballock.pdf

25.For an example of matrices giving gimbal lock, see Justin Wyss-Gallifent’s MATH431 lecture “Gimbal Lock,” November 3, 2021: http://www.math.umd.edu/~immortal/MATH431/book/ch_gimballock.pdf.

26光谱线:如果原子吸收能量,其电子会跃迁到更高的能态;当它们返回到其原始的更稳定状态时,原子会发射光子,该光子显示为彩色光谱线。颜色对应发射光的波长,而波长又与能量跃迁的大小和原子的组成有关。这就是为什么先驱女天文学家安妮·坎农 (Annie Jump Cannon) 自 1896 年以来一直在哈佛大学天文台工作,精心对恒星光谱进行分类以确定其化学成分。她一直这样做,直到 1941 年去世。

26.Spectral lines: If an atom absorbs energy, its electrons jump to a higher energy state; when they return to their original, more stable state, the atom emits a photon, which shows up as a coloured spectral line. The colour corresponds to the emitted light’s wavelength, which in turn is related to the size of the energy jump and the makeup of the atom. Which is why the pioneering female astronomer Annie Jump Cannon had been working at Harvard College Observatory since 1896, painstakingly classifying the spectra of stars to determine their chemical composition. She continued doing this till 1941, the year she died.

27实际上,与 Stern-Gerlach 的联系是稍后才建立起来的。无论如何,Uhlenbeck 和 Goudsmit 的值为 ± h /4π;符号取决于自旋轴是与磁场对齐还是相反,幅度是“标准化”普朗克常数h /2π 的一半(表示为ħ,发音为“h-bar”)。这就是为什么现在说电子的自旋为 ½。

27.Actually, the connection with Stern-Gerlach was made a little later. Anyway, Uhlenbeck and Goudsmit’s values were ±h/4π; the sign depends on whether the spin axis is aligned with or against the magnetic field, and the magnitude is half the “normalised” Planck constant h/2π (which is represented as ħ, pronounced “h-bar”). That’s why the electron is now said to have a spin of ½.

28埃伦费斯特,洛伦兹:古德斯米特在 1971 年 4 月为荷兰物理学会成立 50 周年庆典宣读了一篇精彩的论文,讲述了电子自旋的发现;https://www.lorentz.leidenuniv.nl/history/spin/goudsmit.html

28.Ehrenfest, Lorentz: Goudsmit gave an account of the discovery of electron spin in a delightful paper he read for the golden jubilee of the Dutch Physical Society in April 1971; https://www.lorentz.leidenuniv.nl/history/spin/goudsmit.html.

29泡利矩阵和四元数:泡利矩阵和四元数之间的关系还表明,你可以拥有一个实际上是矩阵的矢量“矢量空间”。重要的是矢量的规则:如果某个东西的行为像矢量,那么就可以将其视为矢量。泡利论四元数:W.泡利,《量子力学的一般原理》,Springer-Verlag,柏林/海德堡,115;他指出,i乘以矩阵遵循单位四元数乘法的规则。狄拉克:PAM狄拉克,《电子的量子理论》,《伦敦皇家学会学报》,A辑,117,778(1928 年 2 月 1 日):610-24;以及保罗·A·M·狄拉克,《电子和正电子理论》,1933 年诺贝尔奖演讲,https://www.nobelprize.org/uploads/2018/06/dirac-lecture.pdf

29.Pauli matrices and quaternions: What the relationship between Pauli matrices and quaternions also shows is that you can have a “vector space” of vectors that are actually matrices. It is the rules of vectors that matter: if something behaves like a vector, it can be treated as one. Pauli on quaternions: W. Pauli, General Principles of Quantum Mechanics, Springer-Verlag, Berlin/Heidelberg, 115; he noted that i times the matrix obeys the rules of unit quaternion multiplication. Dirac: P. A. M. Dirac, “The Quantum Theory of the Electron,” Proceedings of the Royal Society of London, series A, 117, 778 (February 1, 1928): 610–24; and Paul A. M. Dirac, “Theory of Electrons and Positrons,” 1933 Nobel Prize speech, https://www.nobelprize.org/uploads/2018/06/dirac-lecture.pdf.

30四元数和自旋旋转:狄拉克的新理论表明,物质的所有构成要素(电子、质子和中子)的自旋都包含 ½;也就是说,它们是 ½ 的奇数倍,它们被称为费米子。(光子和其他所谓的玻色子具有整数自旋。)

30.Quaternion and spin rotations: Dirac’s new theory showed that all the building blocks of matter—electrons, protons, and neutrons—have spins that contain a ½; that is, they are odd-number multiples of ½, and they are called fermions. (Photons and other so-called bosons have integer spins.)

粒子自旋中的 ½ 导致了奇怪的行为,即需要两次360° 旋转才能将费米子恢复到其原始状态。这一发现来自旋转量子粒子自旋轴所需的数学——实际上类似于四元数旋转。这并不奇怪,因为两者都给出了相同的违反直觉的旋转类型。我在这里给出了一个简短的总结:

It’s the ½ in a particle’s spin that causes the strange behavior in which you need two 360° rotations to bring a fermion back to its original state. This discovery came out of the maths you need to rotate a quantum particle’s spin axis—similar maths to quaternion rotations, as it happens. Which isn’t really surprising, given that each gives the same counterintuitive type of rotation. I’ve given a brief summary here:

正如我在图 4.4和四元数旋转的尾注中所暗示的那样,要将矢量绕x轴旋转角度 θ,单位四元数为=θ2类似地,要将自旋半角动量矢量绕x轴旋转 θ ,需要一个幺正算符=θ2σ,其中单位选择为 2π/ h = 1,σ x为泡利自旋矩阵。注意两种情况下的θ /2 — 这就是为什么需要两次完整的 2π 旋转才能回到原始未旋转状态(θ = 0 + 2 n π)。

As I implied in fig. 4.4 and the endnote on quaternion rotations, to rotate a vector through an angle θ around the x-axis, the unit quaternion is U=eiθ2. Similarly, to rotate a spin half angular momentum vector through θ about the x-axis, you need a unitary operator U=eiθ2σx, where units are chosen so that 2π/h = 1, and σx is a Pauli spin matrix. Note the θ/2 in both cases—it’s why you need two full 2π rotations to get back to the original unrotated (θ = 0 + 2nπ) state.

相比之下,要将轨道而不是自旋角动量绕x轴旋转 θ 角,幺正算符为U = e i θ Jx。旋转 2π 后,您又回到起点。还要注意,在四元数和量子旋转中,您还需要后乘以逆或共轭;这称为“相似变换”。

By contrast, to rotate the orbital rather than the spin angular momentum through an angle θ around the x-axis, the unitary operator is U = eiθ Jx. Rotate through 2π and you are back where you started. Note, too, that in both quaternion and quantum rotations, you also need to postmultiply by the inverse or conjugate; this is called a “similarity transformation.”

描述四元数和自旋之间联系的另一种方式是,四元数和自旋半旋转矩阵都是群SU (2) 的元素。群论处理的是底层结构,因此通过研究各种数学或物理结构的群性质,有时可以发现两个看似非常不同的东西之间的相似之处。

Another way of describing the connection between quaternions and spin is that both quaternion and spin half rotation matrices are elements of the group SU(2). Group theory deals with underlying structures, so by studying group properties of various mathematical or physical structures, you can sometimes spot similarities between two apparently very different things.

31克莱因和奥帕特使用的是中子而不是电子,因为实验中电子上的电荷会对外部磁场产生过强的干扰。他们使用铁磁晶体将一束中子(以物质波的形式传播)衍射成两部分,其中一部分与外部磁场相互作用。(在量子力学中,“物质波”或“波函数”描述的是粒子在特定时间和地点被探测到的概率。)他们发现,当一束中子(波的一部分)旋转 360° 或 2π 弧度的奇数倍时(当然,包括 2π 本身),它会与另一半未旋转的中子发生破坏性干扰,产生独特的干涉图案。对于 2π 的偶数倍,破坏性干扰就会消失。因此,在 2π 的奇数倍情况下,您需要再次旋转自旋,以便它全部经过偶数倍(例如 4π)才能使干涉图案恢复正常。 AG Klein 和 GI Opat,“通过中子的菲涅尔衍射观察 2π 旋转”,物理评论快报37,第 5 期(1976 年 8 月 2 日):238-40。

31.Klein and Opat used neutrons rather than electrons, because the charge on electrons would interfere too strongly with the external magnetic field in the experiment. They used a ferromagnetic crystal to diffract a beam of neutrons (traveling as a matter wave) into two parts, one of which interacted with the external magnetic field. (In quantum mechanics the “matter wave” or “wave function” describes the probability of a particle being detected at a certain time and place.) What they found was that when one beam—one part of the wave—has been rotated by an odd-integer multiple of 360° or 2π radians— which, of course, includes 2π itself—it interfered destructively with the other, nonrotated half, producing a distinctive interference pattern. For even multiples of 2π, the destructive interference disappeared. So in the odd-integer 2π case, you need to rotate the spin again, so that all up it goes through an even multiple such as 4π to get the interference pattern back to normal. A. G. Klein and G. I. Opat, “Observation of 2π Rotations by Fresnel Diffraction of Neutrons,” Physical Review Letters 37, no. 5 (August 2, 1976): 238–40.

32其他团队由 Helmut Rauch 和 Sam Werner 领导,但这三个团队并非竞争对手,而是随后开始合作,正如克莱因在“中子干涉测量:三个大陆的故事”中所描述的那样,可在http://www.europhysicsnews.orghttp://dx.doi.org/10.1051/epn/2009802上找到。

32.The other teams were headed by Helmut Rauch and Sam Werner, but instead of being rivals the three teams subsequently worked together, as Klein describes in “Neutron Interferometry: A Tale of Three Continents,” available at http://www.europhysicsnews.org or http://dx.doi.org/10.1051/epn/2009802.

33布朗在《维多利亚时代科学家的诗歌》第 7-9 页中分析了汉密尔顿的颂歌

33.Hamilton’s Ode is analysed by Brown, Poetry of Victorian Scientists, 7–9.

34当今八元数:有关当前研究的技术概述,请参阅 Peter Rowlands 和 Sydney Rowlands 的“八元数对标准是否必要模型?》,《物理学杂志:会议系列1251,012044(2019 年),DOI 10.1088/1742-6596/1251/1/012044。其中一位研究人员是加拿大年轻女性 Cohl Furey;另一位是加州大学数学家 John Baez。2021 年,Baez 在https://math.ucr.edu/home/baez/standard/上更新了情况。

34.Octonions today: For a technical overview of current research, see Peter Rowlands and Sydney Rowlands, “Are Octonions Necessary to the Standard Model?,” Journal of Physics: Conference Series 1251, 012044 (2019), DOI 10.1088/1742-6596/1251/1/012044. One of these researchers is a young Canadian woman, Cohl Furey; another is University of California mathematician John Baez. In 2021, Baez gave an update on the situation at https://math.ucr.edu/home/baez/standard/.

第五章

CHAPTER 5

1 .罗伯特·P·格雷夫斯(Robert P. Graves),《威廉·罗恩·汉密尔顿爵士生平》,3卷。(都柏林:霍奇斯,菲吉斯,1882 年、1885 年、1889 年),2:585–86。

1.Robert P. Graves, The Life of Sir William Rowan Hamilton, 3 vols. (Dublin: Hodges, Figgis, 1882, 1885, 1889), 2:585–86.

2格雷夫斯,《汉密尔顿传》,2:586。

2.Graves, Life of Hamilton, 2:586.

3 .关于格拉斯曼的早期生活和工作,以及下文的大部分内容,我主要受益于 Michael J. Crowe 的《向量分析史》(印第安纳州圣母大学:圣母大学出版社,1967 年),第 3 章(更多详情请参阅 Hans-Joachim Petsche 的《赫尔曼·格拉斯曼》 [巴塞尔:Birkhäuser,2009 年]);以及 Jean-Luc Dorier 的《向量空间理论起源概述》,《数学史》第 22 卷(1995 年):227-261 页,其中介绍了哈密尔顿和格拉斯曼等向量的发现。

3.For Grassmann’s early life and work here and in much of the following, I’m indebted primarily to Michael J. Crowe, History of Vector Analysis (Notre Dame, IN: University of Notre Dame Press, 1967), chap. 3 (for more detail see Hans-Joachim Petsche, Hermann Grassmann [Basel: Birkhäuser, 2009]); and for an overview of the discovery of vectors, including Hamilton and Grassmann, Jean-Luc Dorier, “A General Outline of the Genesis of Vector Space Theory,” Historia Mathematica 22 (1995): 227–61.

4 .格拉斯曼“震惊”了:1847 年写给圣维南的信,引用自 Crowe的《向量分析史》,56 页。

4.Grassmann “astounded”: 1847 letter to Saint-Venant, quoted in Crowe, History of Vector Analysis, 56.

5Crowe,《矢量分析史》,70-72。

5.Crowe, History of Vector Analysis, 70–72.

6 .狂热的新当局对意大利出生的拉格朗日犹豫不决——外国人通常会被剥夺职位和财产——但最终一项特别法令允许他留任,并担任委员会主席。这在很大程度上要归功于先驱化学家安托万·拉瓦锡为外国科学家所做的努力——他和他的妻子玛丽被称为现代化学的“父亲和母亲”。拉瓦锡因在税务公司的商业利益而悲惨地被送上了断头台,拉格朗日曾说过一句名言,虽然暴徒只花了一瞬间就砍下了他的头,但可能要过一个世纪才能恢复原状。十八个月后,政府宣布拉瓦锡与税务公司没有任何不当行为。

6.The fanatical new authorities had hesitated over Italian-born Lagrange— foreigners were usually stripped of their posts and possessions—but in the end a special decree enabled him to stay on, and as head of the committee, no less. This was largely thanks to the efforts on behalf of foreign scientists by pioneering chemist Antoine Lavoisier—he and his wife Marie have been dubbed the “father and mother” of modern chemistry. Lavoisier famously and tragically ended up at the guillotine for his business interests in a taxcollecting company, and Lagrange famously declared that while it took the mob but a moment to cut off his head, it might take a century to see its like again. Eighteen months later the government declared Lavoisier innocent of any wrongdoing by the tax company.

7一些消息来源称有两个孩子幸存下来:无论如何,这是多么悲惨的事情——尤其是对他可怜的母亲来说!难怪拉格朗日自己没有孩子。

7.Some sources say two children survived: either way, how tragic—especially for his poor mother! No wonder Lagrange himself had no children.

8如果通过向量积将表示定义平行四边形的两条有向线的向量相乘,则会得到另一个向量。(这使得向量积“封闭”。)乘积向量垂直于两个原始向量的平面。但是,如果将通过格拉斯曼的外积计算平行四边形,你得到的不是另一条“线”——格拉斯曼没有使用“矢量”这个术语——而是一个有向面积。因此,外积在概念上不同于矢量积。(不过,正如我们将看到的,它与张量积有关。)但是,它们是等价的,因为你从两边的矢量积得到的矢量与格拉斯曼的有向面积方向相同,大小也相同。

8.If you multiply the vectors representing the two directed lines defining a parallelogram via the vector product, you get another vector. (This makes the vector product “closed.”) The product vector is perpendicular to the plane of the two original vectors. But if you multiply the two sides of the parallelogram via Grassmann’s outer product, you get not another “line”— Grassmann didn’t use the term “vector”—but a directed area. So the outer product is conceptually different from the vector product. (It is related to a tensor product, though, as we’ll see.) However, they are equivalent in that the vector you get from the vector product of the two sides is in the same direction as Grassmann’s directed area and has the same magnitude.

9汉密尔顿写给莫蒂默·奥沙利文的信发表在格雷夫斯的《汉密尔顿传》第 2 卷:683 页。

9.Hamilton’s letter to Mortimer O’Sullivan was published in Graves, Life of Hamilton, 2:683.

10赫歇尔致汉密尔顿,引自格雷夫斯《汉密尔顿传》第 3 卷:121 页。

10.Herschel to Hamilton, quoted in Graves, Life of Hamilton, 3:121.

11Möbius,Apelt,Baltzer,引自 Crowe 的《矢量分析史》,78–80 页。

11.Möbius, Apelt, Baltzer, quoted in Crowe, History of Vector Analysis, 78–80.

12Möbius,Apelt,Baltzer,引自 Crowe 的《矢量分析史》,78–80 页。

12.Möbius, Apelt, Baltzer, quoted in Crowe, History of Vector Analysis, 78–80.

13Möbius,Apelt,Baltzer,引自 Crowe 的《矢量分析史》,78–80 页。

13.Möbius, Apelt, Baltzer, quoted in Crowe, History of Vector Analysis, 78–80.

14安培 vs. 格拉斯曼:若想了解当时的正确评价,请参阅麦克斯韦的《电磁论》,1891 年(原版 1873 年第 3 版,克拉伦登出版社或多佛出版社重印),第 482 页、509–10 页(第 511–25 页为麦克斯韦自己的分析),第 526 页(麦克斯韦倾向于安培,因为格拉斯曼的公式违反了牛顿第三定律),但第 687 页为麦克斯韦的结论,即无法通过实验在这两个公式之间做出决定。

14.Ampère vs. Grassmann: For a contemporary and well-judged assessment, see Maxwell’s Treatise on Electricity and Magnetism, 1891 (3rd edition of the 1873 original, Clarendon Press or Dover reprint), arts. 482, 509–10 (511–25 for Maxwell’s own analysis), 526 (Maxwell favours Ampère because Grassmann’s formula violated Newton’s third law), but 687 for Maxwell’s conclusion that it was impossible to decide experimentally between the two formulae.

最近尝试从实验和概念上做出决定:Christine Blondel 和 Bertrand Wolff,trans。 Andrew Butricia,“安培力定律:一个过时的公式?” Histoire de l'Électricité et du Magnetisme(2009 年 5 月;2013 年翻译,2021 年修订),http://www.ampere.cnrs.fr/histoire/parcours-historique/lois-courants/force-obsolete/eng

Recent attempts to experimentally and conceptually decide: Christine Blondel and Bertrand Wolff, trans. Andrew Butricia, “Ampère’s Force Law: An Obsolete Formula?” Histoire de l’Électricité et du Magnetisme (May 2009; trans. 2013, rev. 2021), http://www.ampere.cnrs.fr/histoire/parcours-historique/lois-courants/force-obsolete/eng.

本文简要概述了近期的研究,不偏不倚。有关详细的、最终支持安培的解释,请参见:AKT Assis 和 JPMC Chaib,《安培电动力学》(Apeiron,2015 年),第 14、16.4 章和结论,第 491 页;不过,1996 年 Assis 已经证明(与 Marcelo A. Bueno 一起)在安培所提倡的实验装置中,格拉斯曼公式和安培公式是等效的:“安培力和格拉斯曼力之间的等效性”,IEEE Transactions on Magnetics 32,第 2 期(1996 年 3 月):第 431-36 页。

This paper gives a brief nonpartisan overview of recent research. For a detailed, ultimately pro-Ampère account: A. K. T. Assis and J. P. M. C. Chaib, Ampère’s Electrodynamics (Apeiron, 2015), chaps. 14, 16.4, and conclusion, 491; still, in 1996 Assis had shown (with Marcelo A. Bueno) that Grassmann’s and Ampère’s formulae were equivalent in the experimental set-up espoused by Ampère: “Equivalence between Ampère and Grassmann’s Forces,” IEEE Transactions on Magnetics 32, no. 2 (March 1996): 431–36.

请注意,一些现代作者正在利用安培与格拉斯曼的争论来质疑场论方法(格拉斯曼的结果在麦克斯韦之后融入其中);参见安培的超距作用。麦克斯韦本人说,拥有多种看待事物的方式总是一个好主意!

Note that some modern authors are using the Ampère vs. Grassmann debate to question the field theory approach (into which Grassmann’s result was integrated post-Maxwell); cf. Ampère’s action-at-a-distance. Maxwell himself said it was always a good idea to have more than one way of seeing things!

Grassmann 的原始论文:Hermann Grassmann,“Neue Theorie der Elektrodynamik”,Annalen der Physik und Chemie 1 (1845):1–18。

Grassmann’s original paper: Hermann Grassmann, “Neue Theorie der Elektrodynamik,” Annalen der Physik und Chemie 1 (1845): 1–18.

15莱布尼茨和格拉斯曼:Joseph Kouneiher,《破缺对称性、无意义空间和莱布尼茨的遗产:物理学的起源》,《理论物理学高级研究》(2015 年 9 月),取自 ResearchGate,https ://www.researchgate.net/publication/281526332_Broken_symmetry_Pointless_Space_and_Leibniz%27s_Legacy_the_origin_of_physics 。

15.Leibniz and Grassmann: Joseph Kouneiher, “Broken Symmetry, Pointless Space and Leibniz’s Legacy: The Origin of Physics,” Advanced Studies in Theoretical Physics (September 2015), accessed from ResearchGate, https://www.researchgate.net/publication/281526332_Broken_symmetry_Pointless_Space_and_Leibniz%27s_Legacy_the_origin_of_physics.

16格雷夫斯,《汉密尔顿传》,3:424。

16.Graves, Life of Hamilton, 3:424.

17格雷夫斯,《汉密尔顿传》,3:441-442。

17.Graves, Life of Hamilton, 3:441–42.

18威廉·罗文·汉密尔顿,《四元数讲座》(伦敦:惠特克和剑桥:麦克米伦,1853 年),例如 59,用于乘以j并改变望远镜的方向。

18.William Rowan Hamilton, Lectures on Quaternions (London: Whittaker, and Cambridge: Macmillan, 1853), e.g., 59, for multiplication by j and changing the orientation of a telescope.

19Crowe 令人信服地证明了向量分析的史学是源自 Hamilton 及其后继者而非 Grassmann(《向量分析史》,77,第 4 章)。我们将看到,很久以后,Grassmann 又影响了 Cartan、Clifford 及其追随者

19.Crowe has made a convincing case for the historiography of vector analysis proceeding from Hamilton and his successors rather than Grassmann (History of Vector Analysis, 77, chap. 4). Much later, as we’ll see, Grassmann will influence Cartan, Clifford, and their followers.

第六章

CHAPTER 6

1 .有关麦克斯韦生活和工作的通俗易懂的描述,请参阅我的《爱因斯坦的英雄:用数学语言想象世界》(圣卢西亚:昆士兰大学出版社,2003 年;纽约:牛津大学出版社,2005 年)。

1.For an accessible account of Maxwell’s life and work, see my Einstein’s Heroes: Imagining the World through the Language of Mathematics (St. Lucia: University of Queensland Press, 2003; New York: Oxford University Press, 2005).

2Tait 激动的信:Cargill Gilston Knott,《PG Tait 的生活和科学著作》(伦敦:剑桥大学出版社,1911 年),第 9 页。关于 Tripos 考试:八天内进行十六次考试的数字是 1854 年(麦克斯韦年):DO Forfar,“高级考试员后来怎么样了?” 《数学谱》第 29 卷,第 1 期(1996 年);可在www.clerkmaxwellfoundation.org上查阅。

2.Tait’s excited letter: Cargill Gilston Knott, The Life and Scientific Work of P. G. Tait (London: Cambridge University Press, 1911), 9. On the Tripos: the figure of sixteen exams over eight days was for 1854 (Maxwell’s year): D. O. Forfar, “What Became of the Senior Wranglers?” Mathematical Spectrum 29, no. 1 (1996); available at www.clerkmaxwellfoundation.org.

3 .麦克斯韦在剑桥:刘易斯·坎贝尔和威廉·加内特,《詹姆斯·克拉克·麦克斯韦的生平》(伦敦:麦克米伦,1882 年),第 94-95 页。(1997 年 Sonnet Software 推出了数字版。)

3.Maxwell at Cambridge: Lewis Campbell and William Garnett, The Life of James Clerk Maxwell (London: Macmillan, 1882), 94–95. (There is a 1997 digital edition by Sonnet Software.)

4 .这首诗的全名是《牧马人、大学、学究和哲学的愿景》,发表于坎贝尔和加内特的《麦克斯韦传》第 307 页。

4.The poem—whose full title is “A Vision of a Wrangler, of a University, of Pedantry, and of Philosophy”—is published in Campbell and Garnett, Life of Maxwell, 307.

5麦克斯韦担任审查员:坎贝尔和加内特,《麦克斯韦生平》,175。

5.Maxwell as examiner: Campbell and Garnett, Life of Maxwell, 175.

6 .泰特谈麦克斯韦:讣告,《爱丁堡皇家学会会刊》第 10 卷(1878–80 年):第 331–39 页。其父亲写给麦克斯韦的信:坎贝尔和加内特, 《麦克斯韦生平》,第 109 页。其老老师:大卫·O·福法和克里斯·普里查德,《麦克斯韦和泰特的非凡故事》,《詹姆斯·克拉克·麦克斯韦纪念册》(爱丁堡,1999 年),第 3 页。

6.Tait on Maxwell: Obituary, Proceedings of the Royal Society of Edinburgh 10 (1878–80): 331–39. His father’s letter to Maxwell: Campbell and Garnett, Life of Maxwell, 109. His old teacher: David O. Forfar and Chris Pritchard, “The Remarkable Story of Maxwell and Tait,” James Clerk Maxwell Commemorative Booklet (Edinburgh, 1999), 3.

7关于斯托克斯定理:麦克斯韦在《电磁论》 (牛津:克拉伦登出版社,1873 年)第 1 章第 27 节中将证明归功于汤姆森和泰特(并指出该定理首次出现在史密斯奖考试中)。汤姆森可能是该定理的创始人,因为他早在 1850 年就将其包含在写给斯托克斯的一封信中。维克多·J·卡茨(Victor J. Katz)(《斯托克斯定理的历史》, 《数学杂志》 [MAA] 52,第 3 期 [1979 年 5 月]:146-56)认为赫尔曼·汉克尔是第一个发表的证明(1861 年),但不如汤姆森的证明(1867 年)那么普遍。

7.On Stokes’s theorem: Maxwell credits proof to Thomson and Tait (and also notes its first appearance in the Smith’s Prize exam) in his A Treatise on Electricity and Magnetism (Oxford: Clarendon Press, 1873), 1:27. Thomson is the theorem’s likely originator, for he had included it in a letter to Stokes back in 1850. Victor J. Katz (“The History of Stokes’ Theorem,” Mathematics Magazine [MAA] 52, no. 3 [May 1979]: 146–56) credits Hermann Hankel with the first published proof (in 1861), but it wasn’t as general as Thomson’s (1867).

8圆面积的曲面积分:想象一个由圆包围的曲面上的微小元素dS ,其思想是将dS在整个曲面上积分以求出面积。这里,x轴和y轴代表定义曲面的两个维度,因此你可以想象每个方向上的小线段dxdy,它们与曲面的矩形元素区域接壤——因此在曲面积分中,你要对dS = dxdy进行积分。(因为曲面是一个平面,所以这实际上只是一个二重积分,而不是曲面积分,其中dS比dxdy更复杂,因为它需要使用矢量来找到曲面的法线。)将其转换为极坐标(如图 2.2a所示),并且不要忘记用于改变坐标的雅可比因子,你将得到dS = dx dy = rdr d θ。围绕半径为R的圆对其进行积分:

8.Area of a circle as a surface integral: imagining a tiny element dS of the surface bounded by the circle, the idea is to integrate dS over the whole surface to find the area. Here the x and y axes represent the two dimensions that define the surface, so you can imagine little line segments in each direction, dx and dy, which border a rectangular elemental area of the surface—so in the surface integral you are integrating with respect to dS = dxdy. (Because the surface is a plane, this is actually just a double integral rather than a surface integral, where dS is more complicated than dxdy because it requires the use of vectors to find the normal to the surface.) Transforming this to polar coordinates (as in fig. 2.2a), and not forgetting the Jacobian factor for changing the coordinates, you get dS = dx dy = rdr dθ. Integrate this around the circle of radius R:

区域=02π0Rrdrdθ=0212R2dθ=πR2

Area=02π0Rrdrdθ=02x12R2dθ=πR2.

如果你不熟悉雅可比矩阵,那么对于极坐标,你可以想象一个圆的扇形,其角度为d θ ,半径为r :扇形的弧长s为rd θ (因为s2πr=dθ2π,根据弧度的定义),径向元素的长度为dr,因此面积元素为rdr d θ。

If the Jacobian is not familiar, for polar coordinates you can think of a tiny sector of a circle with angle dθ and radius r: the arc length s of the sector is rdθ (since s2πr=dθ2π, by definition of a radian), and a radial element has length dr, so the element of area is rdr dθ.

9庄园的保留以及 Glenlair 庄园和附属建筑的修复,很大程度上要归功于庄园现任主人 Duncan Ferguson 船长的努力。我很高兴在 Glenlair 见到了 Duncan,您可以在http://www.glenlair.org.uk上看到更多关于他和 Glenlair 信托基金代表 Maxwell 和 Glenlair 所做的工作。

9.The survival of the grounds, along with the restoration of Glenlair House and outbuildings, is thanks largely to the efforts of the estate’s current owner, Captain Duncan Ferguson. I’ve had the pleasure of meeting Duncan at Glenlair, and you can see more about his and the Glenlair Trust’s work on behalf of Maxwell and Glenlair at http://www.glenlair.org.uk.

10麦克斯韦在英国协会会议上的情景,由威廉·斯旺回忆,载于坎贝尔和加内特所著的《麦克斯韦生平》,第 236 页。

10.Maxwell at British Association meeting, recollected by William Swan, in Campbell and Garnett, Life of Maxwell, 236.

11静止引力?当然,太阳和行星都在运动,但在轨道的任何给定点,它们相对于彼此都是静止的,并且相距给定的距离。牛顿定律处理的是仅由这个距离和两个质量产生的力。库仑定律:麦克斯韦在他的《电磁论》 1:34, 75 中描述了建立平方反比定律的更精确的实验。

11.Stationary gravity? The sun and planets are moving, of course, but at any given point in the orbit they are stationary with respect to each other and are at a given distance apart. Newton’s law treats the force arising just from this distance and the two masses. Coulomb’s law: Maxwell described more accurate experiments establishing the inverse square law in his Treatise on Electricity and Magnetism, 1:34, 75.

12拉格朗日势的(简化)数学公式为:功 = 力乘以距离,因此如果移动的距离在垂直方向(例如y),则W = f × y 。如果力保持不变,则此公式适用,但对于随距离变化的力,例如重力,则需要使用积分微积分来“加总”力乘以力移动物体时每个点的增量距离。

12.The (simplified) maths of Lagrange’s potential: Work = force times distance, so if the distance moved is in the vertical direction (y, say), then W = f × y. This formula is fine if the force remains constant, but for forces that change with distance, such as gravity, you need integral calculus to “add up” the force times the incremental distance at each point as the force moves the object.

牛顿用力曲线下的面积给出了几何微积分的定义,但在莱布尼茨符号中,我们有西=一个bfd,得出F ( b )−F ( a ),其中F是f的不定积分。(实际上,这只对“保守”力成立,包括重力,它们只取决于积分的端点;对于其他力,如摩擦力,则需要线积分,以考虑起点a和终点b之间整个路径的特性。)

Newton had given a geometrical calculus definition of this in terms of the area under the force curve, but in Leibnizian notation we’d have W=abfdy, which gives F(b) − F(a), where F is the antiderivative of f. (Actually, this is true only for “conservative” forces, including gravity, which depend only on the endpoints of the integral; for other forces, such as friction, you need a line integral, to take account of the characteristics of the entire path between the starting point a and finishing point b.)

拉格朗日实际上表明的是f=dFd这只是微积分入门课程中教授的微积分基本定理,但拉格朗日将其扩展到三维,以允许力和移动的距离朝任何方向移动,而不仅仅是上下移动——因此他发现力有分量FFF(“花括号 d” 符号当时还不是标准符号,但我使用它是为了不让现代数学读者感到困惑。)继牛顿之后,他提出了力是矢量的想法,但与汉密尔顿和格拉斯曼之前的所有人一样,他只处理分量,而不是整个矢量。

What Lagrange showed, in effect, was that f=dFdy. This is just the fundamental theorem of calculus taught in introductory calculus classes, but Lagrange extended it to three dimensions, to allow for the force and distance moved to be in any direction, not just up and down—so he found that the force has components Fx,Fy,Fz. (The “curly d” notation wasn’t standard then, but I’m using it so as not to confuse modern mathematical readers.) Following Newton, he had the idea of force as a vectorial quantity, but like everyone before Hamilton and Grassmann, he dealt only with components, not with whole vectors.

根据功和势能之间的关系,F被称为与力f相关的“势” 。通常,力用大写字母F表示,因此势的常用符号是V。(乔治·格林于 1828 年首次使用“势”一词。)

Drawing on the relationship between work and potential energy, F is called the “potential” associated with the force f. Generally force is written with an uppercase F, so a common symbol for potential is V. (George Green was the first to use the term “potential,” in 1828.)

13偏导数:在F项中,F仅对x求导,因此该项表示当yz保持不变时F在x方向上的变化情况,其他两个项亦然。

13.Partial derivatives: in the Fx term, F is differentiated only with respect to x, so this term tells how F changes in the x-direction while y and z remain fixed—and similarly for the other two terms.

14电学中的牛顿”:麦克斯韦,《电磁论》,2:175。

14.Newton of electricity”: Maxwell, Treatise on Electricity and Magnetism, 2:175.

15这种力只取决于端点,而不取决于它们之间路径的性质,被称为“保守力”,因为它们会导致能量守恒。例如,对于牛顿引力,运动是径向的,因此平方反比定律可以写成r¨=r2;写作r¨=r˙dr˙/dr,积分得出将物体从点 1 移动到点 2 时力所做的功:

15.Such forces, which depend only on the endpoints and not on the nature of the path between them, are called “conservative,” because they lead to conservation of energy. For Newtonian gravity, for example, the motion is radial, so the inverse square law can be written as mr¨=GmMr2;; writing r¨=r˙dr˙/dr, integrate to find the work done by the force in moving an object from point 1 to point 2:

r˙1r˙2r˙dr˙=r1r21r2dr12r˙2r=持续的

mr˙1r˙2r˙dr˙=GMmr1r21r2dr12mr˙2GMmr=constant,

这意味着动能和势能的总和是守恒的;常数是从定积分的端点找到的。

which means the sum of the kinetic and potential energies is conserved; the constant is found from the endpoints in the definite integrals.

16麦克斯韦,《法拉第力线讲座》,这是他于 1873 年发表的演讲,收录于他的文集《詹姆斯·克拉克·麦克斯韦的科学书信和论文》,PM Harman 编辑,两卷(剑桥:剑桥大学出版社,1990 年,1995 年),第 803 页。

16.Maxwell, Lecture on Faraday’s Lines of Force, a talk he presented in 1873, in his collected works, The Scientific Letters and Papers of James Clerk Maxwell, ed. P. M. Harman, 2 vols. (Cambridge: Cambridge University Press, 1990, 1995), 803.

17汤姆森和法拉第论场:埃尔南·麦克马林,《物理学中场概念的起源》,《物理学展望》第 4 期(2002 年):第 13-39 页(特别是第 14 页)。本文详细概述了场概念及其演变,直到麦克斯韦提出第一个成熟的场论。

17.Thomson and Faraday on fields: Ernan McMullin, “The Origins of the Field Concept in Physics,” Physics in Perspective 4 (2002): 13–39 (esp. 14). This paper gives a detailed overview of the field concept and its evolution up till Maxwell gave the first full-blown field theory.

18马里歇尔教授职位:Forfar 和 Pritchard(“非凡的故事”,3–4)指出,学院记录虽未存世,但人们认为 Tait 是麦克斯韦所获职位的候选人,后来阿伯丁大学的 John S. Reid 在“詹姆斯·克拉克·麦克斯韦的苏格兰教席”中表示 Tait 是候选人之一,《皇家学会哲学学报 A》(2008 年),366,1661–84,DOI:10.1098/rsta.2007.2177。如果真是这样,如 Forfar 和 Pritchard 所述,麦克斯韦和 Tait 的通信记录表明他们之间并无怨恨,无论是在 1856 年还是 1860 年(当时 Tait 击败麦克斯韦获得爱丁堡职位)。凯莱申请:Crilly,“亚瑟·凯莱:未选择的路”,52。

18.The Marischal professorship: Forfar and Pritchard (“Remarkable Story,” 3–4) noted that College records were not extant, but that it was believed Tait was a candidate for the job that Maxwell got—and later John S. Reid, of the University of Aberdeen, stated that Tait was a candidate, in “James Clerk Maxwell’s Scottish Chair,” Philosophical Transactions of the Royal Society A (2008), 366, 1661–84, DOI:10.1098/rsta.2007.2177. If so, as Forfar and Pritchard note, Maxwell and Tait’s correspondence shows there were no hard feelings between them—in 1856 or in 1860 (when Tait beat Maxwell to a job at Edinburgh). Cayley applying: Crilly, “Arthur Cayley: The Road Not Taken,” 52.

19麦克斯韦解释了他选择的语言:他在 1865 年的论文(J. Clerk Maxwell,《电磁场的动态理论》,《皇家学会哲学学报》伦敦155 [1865]:459–512)开头提到了这一点,并在《电磁论》 1:98–99(第 95 行)和第 2 卷(第 3 版),176–77(第 529 行)中对其进行了充分的解释。他说,普通积分以及有限空间内的线积分和曲面积分适合超距作用方法,而偏微分方程和整个空间的(体积)积分才是场的自然语言。

19.Maxwell explaining his choice of language: He alludes to it at the beginning of his 1865 paper ( J. Clerk Maxwell, “A Dynamical Theory of the Electromagnetic Field,” Philosophical Transactions of the Royal Society London 155 [1865]: 459–512), and explains it fully in his Treatise on Electricity and Magnetism, 1:98–99 (art. 95), and vol. 2 (3rd ed.), 176–77 (art. 529). He says that ordinary integrals, and line and surface integrals over finite spaces, suit the action-at-a-distance approach, while partial differential equations, and (volume) integrals throughout all of space, are the natural language for fields.

20麦克斯韦对电流的定义及其与通量的关系:他包括两种类型的电流:导体中的传统电流,例如线圈,其中电流是电流密度的通量;以及电容器中的有效电流,他称之为“位移电流”,该电流与电容器板之间电力的变化通量成正比。

20.Maxwell’s definitions of current and their relation to flux: He included two types of current: the conventional one in conductors such as a loop of wire—where the current is the flux of the current density—and the effective current in a capacitor, which he called the “displacement current,” and which is proportional to the changing flux of the electric force between the capacitor plates.

21将麦克斯韦场方程中的积分转换为导数:“积分微积分第一基本定理”将积分和导数联系在一起:

21.Converting integrals to derivatives in Maxwell’s field equations: The “first fundamental theorem of integral calculus” links integrals and derivatives:

一个bfd=FbF一个,其中F ( x ) 是f ( x )的不定积分。

abfxdx=FbFa, where F (x)is the antiderivative of f (x).

换句话说,f=dFd,假设相关函数是可积/可微的!这意味着你可以从f ( x ) 到F ( x )通过积分,或者通过微分从F ( x ) 到f ( x )。斯托克斯定理是这一思想的延伸,你可以从(单)线积分到(双)曲面积分,反之亦然。同样,你可以从曲面积分到体积(三重)积分,再通过现在的后向量,即所谓的“散度定理”返回。例如,这是麦克斯韦推导出高斯静电电磁定律微分形式的方式(在他的《电磁论》 1:68、79、98-99):

In other words, fx=dFxdx, assuming the relevant functions are integrable/differentiable! What this means is that you can go from f(x) to F(x) via integration, or from F(x) to f(x) via differentiation. Stokes’s theorem is an extension of this idea, where you can go from (single) line integrals to (double) surface integrals, and vice versa. Similarly, you can go from surface integrals to volume (triple) integrals and back via what is now, post-vectors, known as the “divergence theorem.” For example, this is how Maxwell deduced the differential form of Gauss’s laws for static electricity and magnetism (in his Treatise on Electricity and Magnetism, 1:68, 79, 98–99):

实验得知,一定体积内所含电荷量e可以写成电荷密度ρ的体积积分:

It was known experimentally that the amount of electric charge e contained in a given volume could be written as the volume integral of the charge density ρ:

e = ∫∫∫ρ dx dy dz … 我将此方程称为 (1)。

e = ∫∫∫ρ dx dy dz … I’ll call this equation (1).

我们还知道(根据库仑定律),单位测试电荷对电荷e施加的力R为R = e / r 2,而通过封闭表面的电通量为

It was also known (from Coulomb’s law) that the force R exerted on a charge e by a unit test charge is R = e/r2, and that the electric flux through a closed surface was

∫∫R cos ε dS = 4π e , … (2)

∫∫ R cos ε dS = 4πe, … (2)

其中 ε 是力的方向角。麦克斯韦借鉴法拉第的公式,将R cos ε dS称为通过表面的“感应”。麦克斯韦将R的分量标记为X、Y、Z,并将其与以下定理联系起来(现在称为散度定理,但当时还没有名字,而且只以分量形式为人所知,如图所示):

where ε is the angle of the direction of the force. Maxwell, adapting Faraday, called Rcos εdS the “induction” through the surface. Maxwell labeled the components of R as X, Y, Z, which he linked to the following theorem (now known as the divergence theorem, but it didn’t have a name then, and it was only known in component form as shown):

R余弦εd年代=dd+dd+ddddd… (3)

RcosεdS=dXdx+dYdy+dZdzdxdydz … (3)

因此麦克斯韦将(1)乘以 4π,并将结果与​​ (2)相等,得到

So then Maxwell multiplied (1) by 4π and equated the result with (2), to get

∫∫ R cos ε dS = 4π ∫∫∫ ρ dx dy dz .... (4)

∫∫ R cos ε dS = 4π ∫∫∫ ρ dx dy dz.... (4)

最后,使(3)(4)相等,并将(3)中的闭合曲面作为(4)中体积的一个元素,得到:

Finally, equate (3) and (4), and take the closed surface from (3) as an element of the volume in (4), to get:

dd+dd+dd=4πρ...(5)

dXdx+dYdy+dZdz=4πρ...(5)

如果你已经熟悉矢量微积分,你会认识到左边是R发散,但我们会在 Tait 和 Maxwell(以及 Heaviside)将其转换为矢量形式时在叙述中讨论这一点。

If you are familiar with vector calculus already, you’ll recognise the lefthand side is the divergence of R, but we’ll come to this in the narrative when Tait and Maxwell (and Heaviside) put it into vector form.

同时,正如麦克斯韦随后解释的那样,如果你能把电力写成电位V的形式,那么(5)就变成了拉普拉斯公式的泊松扩展方程。 (5)式的矢量形式是库仑定律在当今麦克斯韦方程中的表示方式。静磁性的结果也类似。

Meantime, as Maxwell then explained, if you can write the electric force in terms of a potential V, then (5) becomes Poisson’s extension of Laplace’s equation. The vector form of (5) is the way Coulomb’s law appears in Maxwell’s equations today. The result for static magnetism follows in a similar way.

我在这里展示的麦克斯韦的工作使通量/面积积分与散度之间的联系变得清晰——它类似于麦克斯韦使用斯托克斯定理将安培定律和法拉第定律表达为微分方程的方式(参见《电磁论》,2:29、45、147-48、233、251、255)。它涉及另一个向量微积分运算,不是散度而是旋度,在下一章中我们将了解这两个向量运算。

Maxwell’s working that I’ve shown here makes the links between flux/surface integrals and divergence clear—and it is analogous to the way Maxwell used Stokes’s theorem to express Ampère’s and Faraday’s laws as differential equations (see Treatise on Electricity and Magnetism, 2:29, 45, 147–48, 233, 251, 255). It involves another vector calculus operation, not divergence but curl, and in the next chapter we’ll meet both these vector operations.

22爱因斯坦的名言出自阿尔伯特·爱因斯坦的《思想与观点》(1954 年;纽约:三河出版社,1982 年),第 327 页。麦克斯韦的“大炮”出自他写给查尔斯·凯的一封信,转载自坎贝尔和加内特的《麦克斯韦传》,第 169 页。

22.Einstein’s quote is from Albert Einstein, Ideas and Opinions (1954; New York: Three Rivers Press, 1982), 327. Maxwell’s great guns is from a letter to Charles Cay, reprinted in Campbell and Garnett, Life of Maxwell, 169.

23这是因为一般的波动方程都是微分的;但要导出电磁波方程,还需要麦克斯韦对安培定律的理论变换,也就是增加“位移电流”。

23.This is because the general wave equation is differential; but to get out the electromagnetic wave equation you also need Maxwell’s theoretical change to Ampère’s law, i.e., the addition of the “displacement current.”

24引自 Anne van Weerden,《维多利亚时代的婚姻:William Rowan Hamilton 爵士》(Stedum,荷兰:J. Fransje van Weerden,2017),10, 56, 326。

24.Quotes are from Anne van Weerden, A Victorian Marriage: Sir William Rowan Hamilton (Stedum, Netherlands: J. Fransje van Weerden, 2017), 10, 56, 326.

25范维尔登,《维多利亚时代的婚姻》,326。

25.Van Weerden, A Victorian Marriage, 326.

26关于爱丁堡的任职(和巴里):Forfar 和 Pritchard,《非凡的故事》。引自巴里:雷蒙德·弗勒德,《汤姆森和泰特:自然哲学论文》,收录于雷蒙德·弗勒德、马克·麦卡特尼和安德鲁·惠特克合著的《开尔文:生平、劳动和遗产》,牛津奖学金在线(2008 年 5 月):DOI:10.1093/acprof:oso/9780199231256.001.0001。吉尔关于麦克斯韦教学的评论:里德,《麦克斯韦的苏格兰讲席教授》,1673 年。

26.On the Edinburgh posting (and Barrie): Forfar and Pritchard, “Remarkable Story.” Quote from Barrie: Raymond Flood, “Thomson and Tait: The Treatise on Natural Philosophy,” in Raymond Flood, Mark McCartney, and Andrew Whitaker, Kelvin: Life, Labours and Legacy, Oxford Scholarship Online (May 2008): DOI: 10.1093/acprof:oso/9780199231256.001.0001. Gill on Maxwell’s teaching: Reid, “Maxwell’s Scottish Chair,” 1673.

第七章

CHAPTER 7

1 .Tait 至 Thomson:收录于 R. Flood,《Thomson 和 Tait:自然哲学论述》,收录于 Raymond Flood、Mark McCartney 和 Andrew Whitaker合著的《开尔文:人生、劳动和遗产》(牛津:牛津大学出版社,2008 年)和 Scholarship Online(2021 年),176。DOI:10.1093/acprof:oso/9780199231 256.003.0011。

1.Tait to Thomson: in R. Flood, “Thomson and Tait: The Treatise on Natural Philosophy,” in Raymond Flood, Mark McCartney, and Andrew Whitaker, Kelvin: Life, Labours and Legacy (Oxford: Oxford University Press, 2008) and Scholarship Online (2021), 176. DOI: 10.1093/acprof:oso/9780199231 256.003.0011.

2Tait 的研究和清单:Cargill Gilston Knott,《P. G. Tait 的生活和科学工作》(伦敦:剑桥大学出版社,1911 年),33、43。

2.Tait’s study and list: Cargill Gilston Knott, The Life and Scientific Work of P. G. Tait (London: Cambridge University Press, 1911), 33, 43.

3 .关于麦克斯韦绰号的由来:该方程出现在 Tait 的《热力学概论》(爱丁堡:埃德蒙斯顿和道格拉斯,1868 年)第 162 节中。另请参阅 MJ Klein 的《麦克斯韦、他的恶魔和热力学第二定律》,《麦克斯韦的恶魔:熵、信息、计算》,哈维·莱夫和安德鲁·雷克斯编(新泽西州普林斯顿:普林斯顿大学出版社,1990 年),第 85–86 页。克莱因的文章发表于 1970 年;麦克斯韦的“妖怪”是一个思想实验,有助于阐明热力学的本质。

3.On the origins of Maxwell’s nickname: The equation appears in section 162 of Tait’s Sketch of Thermodynamics (Edinburgh: Edmonston and Douglas, 1868). See also M. J. Klein, “Maxwell, His Demon, and the Second Law of Thermodynamics,” in Maxwell’s Demon: Entropy, Information, Computing, ed. Harvey Leff and Andrew Rex (Princeton, NJ: Princeton University Press, 1990), 85–86. Klein’s article is from 1970; Maxwell’s “demon” was a thought experiment that helped clarify the nature of thermodynamics.

4 .《麦克斯韦评论詹姆斯·克拉克·麦克斯韦的科学论文》,WD Niven 主编(剑桥:剑桥大学出版社,1890 年),第 326–27 页。麦克斯韦致泰特的信,1871 年 12 月 21 日,诺特,《泰特生平》,第 150 页。

4.Maxwell’s review: Scientific Papers of James Clerk Maxwell, ed. W. D. Niven (Cambridge: Cambridge University Press, 1890), 326–27. Maxwell to Tait, December 21, 1871, Knott, Life of Tait, 150.

5麦克斯韦写给泰特的信(重点是我加的),1870 年 11 月 14 日,载于迈克尔·J·克罗所著的《矢量分析史》(印第安纳州圣母大学:圣母大学出版社,1967 年出版),第 132 页;麦克斯韦写给坎贝尔的信,1872 年 10 月 19 日,载于刘易斯·坎贝尔和威廉·加内特所著的《詹姆斯·克拉克·麦克斯韦的生平》(伦敦:麦克米伦出版社,1882 年出版),第 186 页。麦克斯韦的“分类”论文发表在《伦敦数学学会会刊》(1871 年 3 月 9 日):第 224-33 页。

5.Maxwell to Tait (with my emphasis), November 14, 1870, in Michael J. Crowe, A History of Vector Analysis (Notre Dame, IN: University of Notre Dame Press, 1967), 132; Maxwell to Campbell, October 19, 1872, in Lewis Campbell and William Garnett, The Life of James Clerk Maxwell (London: Macmillan, 1882), 186. Maxwell’s “Classification” paper was published in Proceedings of the London Mathematical Society (March 9, 1871): 224–33.

6 .麦克斯韦给泰特关于矢量微积分名称的信件:1870 年 11 月 7 日,引用自 Knott 的《泰特传》,167 页。

6.Maxwell to Tait about vector calculus names: November 7, 1870, quoted in Knott, Life of Tait, 167.

7麦克斯韦在《电磁论》(牛津:克拉伦登出版社,1873 年)第 1 卷第 28 行(第 25 节)中定义了“收敛”,即发散的负数。叙述中的整个矢量方程在《麦克斯韦论》第 2 卷(第 3 版)第 252、259 行给出。除了减号之外,当您查看第 77 行(或我的图7.1 )和第 612 行中的组件版本时,您会发现他的版本与我们的相同。麦克斯韦在 𝔇 的定义中有一个额外的比例常数K ,但他指出对于空气K = 1。因此,为了将矢量思想放在首位,我通常会假设选择的单位使各种电和磁常数等于 1 来写他的方程。(一些现代文本也在 div E方程中按比例缩小了 4π 。)𝔇 是电位移;叙述中给出的麦克斯韦定义适用于各向同性物质。请注意,在第 2 卷的整个矢量方程中,他使用e而不是 ρ,后者是他在第 1 卷第 77 篇文章中使用的,并且至今仍在使用——所以我在叙述中使用了 ρ。

7.Maxwell defined “convergence,” the negative of divergence, in his Treatise on Electricity and Magnetism (Oxford: Clarendon Press, 1873), 1:28 (art. 25). The whole-vector equations in the narrative are given in Maxwell’s Treatise, vol. 2 (3rd ed.), 252, 259. Minus sign aside, you can see that his version is the same as ours when you look at his component versions, in articles 77 (or my fig. 7.1) and 612. Maxwell has an additional proportionality constant K in his definition of 𝔇, but he notes that for air K = 1. So, to keep the vector ideas foremost, I’ll generally write his equations assuming units are chosen to make various electrical and magnetic constants equal to 1. (Some modern texts have also scaled out the 4π in the div E equation.) 𝔇 is the electric displacement; Maxwell’s definition given in the narrative is for isotropic substances. Note that in his whole vector equation in vol. 2 he uses e instead of ρ, which he’d used in article 77 of vol. 1, and which is used today—so I’ve used ρ in my narrative.

8麦克斯韦致泰特(重点是我加的),1871 年 11 月 2 日,载于克罗著《向量史》,第 133 页。

8.Maxwell to Tait (with my emphasis), November 2, 1871, in Crowe, History of Vectors, 133.

9电磁学和广义相对论中的类似方程包括比安奇恒等式,我们将在第 13 章中简要介绍。在电磁学中,这些恒等式包括 div B方程,反映不存在磁单极子。Tony 和我尚未写出我们的部分结果,但早期的一篇论文是 R. Arianrhod、AW-C. Lun、CBG McIntosh 和 Z. Perjés 的“磁曲率”,《古典与量子引力》第 11 卷(1994 年):2331-34 页。

9.The analogous equations in electromagnetism and general relativity include the Bianchi identities, which we’ll meet briefly in chapter 13. In electromagnetism, these identities include the div B equation reflecting that there are no magnetic monopoles. Tony and I have not yet written up our partial results, but an early paper is R. Arianrhod, A. W.-C. Lun, C. B. G. McIntosh, and Z. Perjés, “Magnetic Curvatures,” Classical and Quantum Gravity 11 (1994): 2331–34.

10麦克斯韦在他的《数学论文集》 2:248(第 604 节)中以分量形式写下了(我们称之为)div B方程,并指出它遵循方程(A)第 591 节(233)。布鲁斯·亨特(《麦克斯韦学派》 [纽约州伊萨卡:康奈尔大学出版社,1991],245)提到了分量形式(亨特将其标记为 A′),但表示麦克斯韦将其写为S 。∇𝔅 = 0。这当然是 248 上分量方程的哈密顿符号等价形式,尽管我在《人本论》第三版中找不到它。

10.Maxwell writes (what we would call) the div B equation in component form in his Treatise, 2:248 (art. 604), noting it follows from equation (A) art. 591 (233). Bruce Hunt (The Maxwellians [Ithaca, NY: Cornell University Press, 1991], 245) mentions the component form (which Hunt labels A′) but says that Maxwell wrote this as S. ∇𝔅 = 0. This is certainly the equivalent in Hamiltonian notation of the component equation on 248, although I can’t find it in my third edition copy of the Treatise.

11矢量势:麦克斯韦,《引力与磁通量》,第 2 卷,第 405、422-23、592 节:他定义了矢量势A ,使得A的线积分等于(通过斯托克斯定理)磁场B的面积分(即矢量势A的旋度)。他还给出了它的物理解释,即磁矩(第 405 节)和电磁动量(第 590、592、618 节)——尽管这是通过数学类比而不​​是直接的物理匹配;例如,他选择了“电磁动量”这个术语,因为从数学上讲它是力的时间积分(第 590 节)——也就是说,它的时间导数是一种力,就像普通的牛顿动量一样。

11.Vector potential: Maxwell, Treatise, vol. 2, arts. 405, 422–23, 592: he defines the vector potential A so that the line integral of A equals (via Stokes’s theorem) the surface integral of the magnetic field B (which is the curl of the vector potential A). He also gives it a physical interpretation, in terms of magnetic moments (art. 405) and electromagnetic momentum, arts. 590, 592, 618—although this is via mathematical analogy rather than by a direct physical match; for example, he chooses the term “electromagnetic momentum” because mathematically it is the time integral of a force (art. 590)—that is, its time-derivative is a force, just like ordinary Newtonian momentum.

12全矢量方程《电磁论》,第 2 卷(第 3 版),第 258 页(分量形式,第 233 页,第 248 页)。:现代符号各不相同,但A 的使用相当广泛,例如 Luciano Maiani 和 Omar Benhar 的《相对论量子力学》(佛罗里达州博卡拉顿:CRC Press,2016 年),第 56 页;Ray D'Inverno 的《爱因斯坦相对论导论》(牛津:Clarendon Press,1992 年),第 160 页;Walter Strauss 的《偏微分方程》(纽约:Wiley,1992 年),第 342 页;Bernard Schutz 的《广义相对论入门课程》(剑桥:剑桥大学出版社,1985 年),第 211 页。

12.The whole-vector equation: Treatise on Electricity and Magnetism, vol. 2 (3rd ed.), 258 (component form, 233, 248). Potential: modern notation varies, but A is used fairly widely, e.g., Luciano Maiani and Omar Benhar, Relativistic Quantum Mechanics (Boca Raton, FL: CRC Press, 2016), 56; Ray D’Inverno, Introducing Einstein’s Relativity (Oxford: Clarendon Press, 1992), 160; Walter Strauss, Partial Differential Equations (New York: Wiley, 1992), 342; Bernard Schutz, A First Course in General Relativity (Cambridge: Cambridge University Press, 1985), 211.

13麦克斯韦致坎贝尔:坎贝尔和加内特,《麦克斯韦生平》,186。

13.Maxwell to Campbell: Campbell and Garnett, Life of Maxwell, 186.

14麦克斯韦的瓦特讲座詹姆斯·克拉克·麦克斯韦的科学书信和论文,PM Harman 编辑,两卷(剑桥:剑桥大学出版社,1990 年,1995 年),791。

14.Maxwell’s Watt Lecture: The Scientific Letters and Papers of James Clerk Maxwell, ed. P. M. Harman, 2 vols. (Cambridge: Cambridge University Press, 1990, 1995), 791.

15Tait 的书《四元数导论》是与他的前任老师 Philip Kelland 合作撰写的。Maxwell的评论(重点是我加的):《自然》第 9 卷(1873 年):137–38 页;Crowe,《矢量史》,133 页。牛顿:致哈雷的信,例如,Nicolae Sfetcu,《艾萨克·牛顿与罗伯特·胡克论万有引力定律》,SetThings(2019 年 1 月 14 日),多媒体出版,DOI:10.13140/RG.2.2.19370.26567,知识共享。

15.Tait’s book, Introduction to Quaternions, was coauthored with his former teacher Philip Kelland. Maxwell’s review (with my emphasis): Nature 9 (1873): 137–38; Crowe, History of Vectors, 133. Newton: letter to Halley, in, e.g., Nicolae Sfetcu, “Isaac Newton vs Robert Hooke on the Law of Universal Gravitation,” SetThings ( January 14, 2019), MultiMedia Publishing, DOI:10.13140/RG.2.2.19370.26567, Creative Commons.

16麦克斯韦、泰特和贝尔福:诺特,泰特的一生,149-50。Kovalevsky:Sophie Kowalevski,“Sur le problème de la spin d'un corps Solide autour d'un point fixe”,Acta Mathematica 12(1889 年 1 月):177-232。它于 1888 年赢得了 Bordin 奖。

16.Maxwell, Tait, and Balfour: Knott, Life of Tait, 149–50. Kovalevsky: Sophie Kowalevski, “Sur le problème de la rotation d’un corps solide autour d’un point fixe,” Acta Mathematica 12 ( January 1889): 177–232. It won the Prix Bordin in 1888.

17麦克斯韦的“物理推理”《电磁论》 1:9(第11条)。

17.Maxwell’s “physical reasoning”: Treatise on Electricity and Magnetism 1:9 (art. 11).

18汤姆森致 RB 海沃德,1892 年,载于克罗著《向量史》,120 页。

18.Thomson to R. B. Hayward, 1892, in Crowe, History of Vectors, 120.

19凯莱的肖像/麦克斯韦的诗:亚历山大·麦克法兰,《十位英国数学家讲座》(1916 年),第 5 章克利福德的体操:卡尔·博耶,《数学史》,修订者乌塔·默茨巴赫(纽约:约翰·威利父子公司,1991 年),第 592 页;另请参阅 Monty Chisholm,“科学与文学的联系:威廉和露西·克利福德的故事”,《应用克利福德代数进展》 19(2009 年):第 657–71 页。

19.Cayley’s portrait/Maxwell’s poem: Alexander MacFarlane, Lectures on Ten British Mathematicians (1916), chap. 5. Clifford’s gymnastics: Carl Boyer, A History of Mathematics, rev. Uta Merzbach (New York: John Wiley and Sons, 1991), 592; see also Monty Chisholm, “Science and Literature Linked: The Story of William and Lucy Clifford,” Advances in Applied Clifford Algebras 19 (2009): 657–71.

20克利福德的无神论:莎莉·沙特尔沃思,《科学与期刊:动物本能与低语机器》,载朱丽叶·约翰编辑,《牛津维多利亚文学文化手册》(2016 年),DOI:10.1093./oxfordhb/9780199593736.013.31。

20.Clifford’s atheism: Sally Shuttleworth, “Science and Periodicals: Animal Instinct and Whispering Machines,” in Juliet John, ed., The Oxford Handbook of Victorian Literary Culture (2016), DOI: 10.1093./oxfordhb/9780199593736.013.31.

21Tait 的评论:转载自 Knott 的《Tait 的生活》,270–72。

21.Tait’s review: reprinted in Knott, Life of Tait, 270–72.

22四元数 q 的逆q −1 = q * / qq *,其中q *是q的复共轭。如果你乘以qq −1 ,你会发现它确实等于 1。

22.Inverse of a quaternion q is q−1 = q*/qq*, where q* is the complex conjugate of q. If you multiply out qq−1 you’ll find that it does indeed give 1.

23美国数学家 David Hestenes 是第一位认识到 Clifford 和 Grassmann 对几何代数的重要性的现代数学家,他在 20 世纪 60 年代就认识到了这一点,此后 Hestenes 和其他人进一步发展了几何代数。楔积在现代张量分析中非常重要(楔积的定义见第 11 章)。

23.The American mathematician David Hestenes was the first modern mathematician to recognise, in the 1960s, the importance of Clifford and Grassmann for geometric algebra, which Hestenes and others have since developed further. Wedge products are important in modern tensor analysis (they are defined in terms of the tensor products we’ll see in chap. 11).

24泰特致凯利:诺特,《泰特生平》,155。

24.Tait to Cayley: Knott, Life of Tait, 155.

25自由功利主义者:David Weinstein,《赫伯特·斯宾塞》,摘自 Edward Zalta 主编的《斯坦福哲学百科全书》(2019 年秋季),plato.stanford.edu。MaxwellTait:例如 Knott,《Tait 的一生》,284–88。

25.Liberal utilitarian: David Weinstein, “Herbert Spencer,” in Edward Zalta, ed., Stanford Encyclopedia of Philosophy (Fall 2019), plato.stanford.edu. Maxwell and Tait: e.g., Knott, Life of Tait, 284–88.

26当今刘易斯的身心分析:Elfed Huw Price,《乔治·亨利·刘易斯(1817-1878):具身认知、活力论和符号感知的进化》,载 Chris Smith 和 Harry Whitaker 主编的《神经科学史上的大脑、心智和意识》(纽约:Springer,2014 年),第 105-23 页。Tait ,刘易斯,布莱克伍德:Gordon Haight 主编的《乔治·艾略特书信集》 (康涅狄格州纽黑文:耶鲁大学出版社,1955 年),第 5 卷,第 401 页、第 417 页和第 9 页第 181 页。

26.Lewes’s mind-body analysis today: Elfed Huw Price, “George Henry Lewes (1817–1878): Embodied Cognition, Vitalism, and the Evolution of Symbolic Perception,” in Brain, Mind and Consciousness in the History of Neuroscience, ed. Chris Smith and Harry Whitaker (New York: Springer, 2014), 105–23. Tait, Lewes, Blackwood: Gordon Haight, ed., The George Eliot Letters (New Haven, CT: Yale University Press, 1955), 5:401, 417, and 9n181.

27麦克斯韦的诗《英国协会,1874》发表在 1874 年 12 月的《布莱克伍德:坎贝尔和加内特:麦克斯韦生平》第 8 页(诗歌重印,第 326 页)。

27.Maxwell’s poem: “British Association, 1874” was published in the December 1874 issue of Blackwood’s: Campbell and Garnett: Life of Maxwell, 8 (poem reprinted, 326).

28泰特和布莱克伍德打高尔夫(和麦克斯韦“更好的一半”):马丁·戈德曼,《以太中的恶魔》(爱丁堡:保罗哈里斯出版社,1983 年),105。

28.Tait and Blackwood at Golf (and Maxwell “better ½”): Martin Goldman, The Demon in the Aether (Edinburgh: Paul Harris Publishing, 1983), 105.

29有关麦克斯韦的诗歌及相关辩论的更多信息,请参阅雷蒙德·弗勒德、马克·麦卡特尼和安德鲁·惠特克合著的《詹姆斯·克拉克·麦克斯韦:他的生平和工作观点》(牛津:牛津大学出版社,2014 年)。

29.For more on Maxwell’s poem and related debates, see Raymond Flood, Mark McCartney, and Andrew Whitaker, James Clerk Maxwell: Perspectives on His Life and Work (Oxford: Oxford University Press, 2014).

30麦克斯韦致坎贝尔:坎贝尔和加内特,《麦克斯韦生平》,202。

30.Maxwell to Campbell: Campbell and Garnett, Life of Maxwell, 202.

31诺特,《泰特传》,261。

31.Knott, Life of Tait, 261.

32肖伯纳致露西:引自奇泽姆,《科学与文学的联系》,第 668 页。

32.Shaw to Lucy: quoted in Chisholm, “Science and Literature Linked,” 668.

第八章

CHAPTER 8

1 .布鲁斯·亨特 (Bruce Hunt) 在他的《麦克斯韦学派》( The Maxwellians ) (纽约州伊萨卡:康奈尔大学出版社,1991 年)中创造了“麦克斯韦学派”一词。洛奇独家报道:詹姆斯·劳蒂奥 (James Rautio),《二十三年:麦克斯韦理论的接受》,《应用计算电磁学学会期刊》第 25 卷,第 12 期(2010 年 12 月),第 998–1006 页。

1.Bruce Hunt coined the term “Maxwellians,” in his The Maxwellians (Ithaca, NY: Cornell University Press, 1991). Lodge scooped: James Rautio, “Twentythree Years: Acceptance of Maxwell’s Theory,” Applied Computational Electromagnetics Society Journal 25, no. 12 (December 2010), 998–1006.

2关于亥维赛:我在此处以及以下段落中引用了 Bruce Hunt 的《奥利弗·亥维赛:一流的奇人》,《今日物理》第 65 卷,第 11 期(2012 年):48–54,DOI:10.1063/PT.3.1788;Jed Buchwald 的《奥利弗·亥维赛,麦克斯韦的使徒和麦克斯韦的叛教者》,《半人马座》第 28(1985 年):288–330;I. Yavetz 的《从晦涩到谜团:奥利弗·亥维赛的作品,1872–1889》(巴塞尔:施普林格出版社,2011 年);以及亥维赛的论文,我通常会在文中引用这些论文。

2.On Heaviside: Here and in the following paragraphs I’ve drawn on Bruce Hunt, “Oliver Heaviside: A First-Rate Oddity,” Physics Today 65, no. 11 (2012): 48–54, DOI: 10.1063/PT.3.1788; Jed Buchwald, “Oliver Heaviside, Maxwell’s Apostle and Maxwellian Apostate,” Centaurus 28 (1985): 288–330; I. Yavetz, From Obscurity to Enigma: The Work of Oliver Heaviside, 1872–1889 (Basel: Springer, 2011); and Heaviside’s papers, which I’ll generally cite as I go.

3 .Maxwell 对 Heaviside 的引用被添加到他第一卷的勘误表(第 2 页)中,参考第 404 页。Heaviside论 Maxwell 的论文:Rautio,“二十三年”。

3.Maxwell’s Reference to Heaviside was added to his list of errata (p. 2) for vol. 1, with reference to p. 404. Heaviside on Maxwell’s Treatise: Rautio, “Twentythree Years.”

4 .天赐的麦克斯韦:奥利弗·亥维赛德,《电磁理论》(伦敦,1893 年;纽约:切尔西出版社,1971 年),1:14。

4.Heaven-sent Maxwell: Oliver Heaviside, Electromagnetic Theory (London, 1893; New York: Chelsea Publishing, 1971), 1:14.

5崇拜四元数:Heaviside,电磁理论,1:136。

5.Worshipping quaternions: Heaviside, Electromagnetic Theory, 1:136.

6 .Heaviside,电磁理论,1:137,139。

6.Heaviside, Electromagnetic Theory, 1:137, 139.

7海维赛德从矢量中消除虚数: 《电磁理论》,1:137,142,149。他的滑稽作品《电磁理论》,1:135。

7.Heaviside eliminating imaginary numbers from vectors: Electromagnetic Theory, 1:137, 142, 149. His drollery: Electromagnetic Theory, 1:135.

8Heaviside 遗漏于 BAAS: Engineering 46(1888):352;引自 Hunt 的“Oliver Heaviside”,52–53。

8.Heaviside missed at BAAS: Engineering 46 (1888): 352; cited in Hunt, “Oliver Heaviside,” 52–53.

9锅和水壶:海维赛德,《电磁理论》,1:203;谋杀潜力:致菲茨杰拉德的信,引自劳蒂奥,《二十三年》,1004。

9.Pot and kettle: Heaviside, Electromagnetic Theory, 1:203; Murdering potentials: letter to FitzGerald, quoted in Rautio, “Twenty-three Years,” 1004.

10海维赛德的形式并不十分“现代”:在大多数本科教科书中,麦克斯韦方程组都以EB 的形式写出,但对于海维赛德来说,EH是单独的。麦克斯韦区分了磁感应或磁场B(一种通量)和磁力H 当磁场完全由磁力引起时,B = μ H,其中 μ 是磁介电常数的系数。这是海维赛德使用的定义,因此在查看他的方程时,可以直接从H改为B 他还使用了麦克斯韦对电位移D (一种通量)的定义,即用力E来表示,即D = c E /4π。

10.Heaviside not quite in “modern” form: In most undergrad textbooks, Maxwell’s equations are written in terms of E and B, but for Heaviside it is E and H that are singled out. Maxwell had distinguished between the magnetic induction or magnetic field, B, which is a flux, and the magnetic force, H. When the magnetic field is induced entirely by the magnetic force, then B = μH, where μ is the coefficient of the magnetic permittivity. This is the definition Heaviside used, so it is straightforward to change from H to B when looking at his equations. He also used Maxwell’s definition of the electric displacement D, a flux, in terms of the force E, namely D = cE/4π.

如今,一些作者仍在使用H,但使用B使得海维赛德方程更加对称。一些作者还效仿麦克斯韦和海维赛德,在散度方程中使用D代替E ,但正如我所提到的,它在数值上与E成正比。

Today some authors still use H, but using B makes Heaviside’s equations more symmetrical. Some authors also use D instead of E in the divergence equation, following Maxwell and Heaviside, but as I mentioned it is numerically proportional to E.

11非物理势能(对于局部能量而言):Buchwald 在《Oliver Heaviside》第 293 页对此作了详细解释。麦克斯韦波动方程:就势能而言, 《论文集》第 2 卷,第 434 页(第 784 页);就磁场而言,《关于光的电磁理论的注释》,《皇家学会哲学学报》第 158 卷(1868 年):第 643-57 页,特别是第 655 页。

11.Potentials unphysical (for localized energy): This is explained in detail in Buchwald, “Oliver Heaviside,” 293. Maxwell’s wave equations: in terms of potential, Treatise 2, 434 (art. 784); in terms of magnetic field, “A Note on the Electromagnetic Theory of Light,” Philosophical Transactions of the Royal Society 158 (1868): 643–57, esp. 655.

在这篇 1868 年的论文中,麦克斯韦的四个场方程并不完全是四个现代方程,但他从中推导出磁场的波动方程。这正是海维赛德在“杀死”势时试图做的!遗憾的是麦克斯韦没有进一步发展这种方法——但正如海维赛德指出的那样(《电磁理论》 ,1:69),麦克斯韦并没有在《人性论》中公正地对待自己的理论:相反,他对迄今为止对电磁学研究的所有贡献进行了精彩的概述。他确实展示了他如何以及为何从早期已知的工作中发展出他的理论,以及它与其他人超距作用模型的比较,但他绝对不是一个自我吹捧者。

In this 1868 paper, Maxwell’s four field equations are not quite the four modern equations, but he deduces from them the wave equation for the magnetic field. This is just what Heaviside was trying to do when he “murdered” the potentials! It’s a pity Maxwell didn’t take this approach further— but as Heaviside noted (Electromagnetic Theory, 1:69), Maxwell didn’t do his own theory justice in the Treatise: instead he provided a brilliant overview of all the contributions to the study of electromagnetism that had been made so far. He certainly showed how and why he developed his theory from the earlier known work, and how it compared with others’ action-at-a-distance models, but he definitely wasn’t a self-promoter.

12这四个方程在不同文本中看起来略有不同;这取决于如何选择电常数和磁常数的单位。具体来说,除了电常数和磁常数外,光速c通常设置为 1(就像我在这里所做的那样),但是,根据海维赛德的说法,通过调整常数的单位,4π 因子通常有效地设置为 1。(此外,一些文本使用海维赛德的“div”和“curl”而不是吉布斯的点和叉。)

12.These four equations look slightly different in different texts; it depends on how the units of the electrical and magnetic constants are chosen. In particular, along with electric and magnetic constants the speed of light c is often set to 1 (as I’ve done here), but, following Heaviside, the factor of 4π is often effectively set to 1 by adjusting the units of the constants. (Also, some texts use Heaviside’s “div” and “curl” instead of Gibbs’s dot and cross.)

13海维赛德对四个关键的麦克斯韦方程中电场和磁场之间的对称性非常着迷,以至于他在方程 ∇ ∙ B = 0 中添加了一个虚构的磁“电荷”(从而假设存在单极子——就像狄拉克在近半个世纪后所做的那样),并在 ∇ × E方程中添加了一个磁“电流”。但这些添加目前还没有任何已知的物理基础(除了人工产生的短寿命量子单极子),因此它们通常被排除在现代电磁方程之外;这意味着——除了矢量形式——现代方程确实是麦克斯韦方程,正如我的叙述所示。

13.Heaviside was so entranced by the symmetry between the electric and magnetic fields in the four key Maxwell equations that he added a fictitious magnetic “charge” to the equation ∇ ∙ B = 0 (thereby positing that there are magnetic monopoles—just as Dirac did nearly half a century later) and a magnetic “current” to the ∇ × E equation. But these additions don’t yet have any known physical basis (aside from artificially generated short-lived quantum monopoles), so they are generally left out of the modern electromagnetic equations; this means that—vector formalism aside—the modern equations are indeed Maxwell’s equations, as my narrative shows.

14从麦克斯韦在《电磁论》第二卷(第 3 版,1891 年;重印,多佛,1954 年)中的整个矢量方程中查找∇ × E:例如,在 2:232(第 590 节)中,麦克斯韦给出了(用文字)定义A = ∫ E dt,或等价地(1865 年论文中的第 29 式)E = − d A / dt。取该式两边的旋度,并记住麦克斯韦定义B = ∇ × A,则可得到(利用可以交换导数顺序的事实)

14.Finding ∇ × E from Maxwell’s whole vector equation in his Treatise on Electricity and Magnetism, vol. 2 (3rd ed., 1891; reprint, Dover, 1954): For example, on 2:232 (art. 590), Maxwell gives (in words) the definition A = ∫E dt, or equivalently (equation 29 of his 1865 paper), E = −dA/dt. Taking the curl of both sides of this, and remembering that Maxwell defined B = ∇ × A, you have (using the fact that you can interchange the order of derivatives)

×=dd×一个=dd

×E=ddt×A=dBdt.

或者,从我的叙述中给出的麦克斯韦方程(论文2:258 [第 619 条])开始,

Alternatively, begin with Maxwell’s equation (Treatise 2:258 [art. 619]) as given in my narrative,

=×d一个dφ

E=v×BdAdtϕ,

然后取两边的旋度。使用旋度梯度 = 0 恒等式,最后一个项消失。如果没有移动电荷,那么电场仅由时变磁场感应产生(参见《论纲》 2:240–41 [艺术 599],2:433 [艺术 783]),则v = 0。因此,你再次×=dd×一个=dd

and then take the curl of both sides. Using the identity curl grad = 0, the last term drops out. If there are no moving charges, so the electric field is induced only by a time-varying magnetic field (cf. Treatise 2:240–41 [art. 599], 2:433 [art. 783]), then v = 0. So again you ×E=ddt×A=dBdt

15麦克斯韦的五个矢量(四元数)方程式见于他的《论著》第 2 卷:258-259 页。(这些页面上还有七个定义方程式——与海维赛德使用的一样。)海维赛德特别指出,他提出的方程式仍应称为麦克斯韦方程式:《电磁理论》,第 1 卷,前言(第 5 页)和第 69 页。赫兹同意:劳蒂奥,《二十三年》,第 1005 页。

15.Maxwell’s five vector (quaternion) equations are in his Treatise 2:258–59. (There are also seven definition equations on these pages—just as Heaviside used.) Heaviside specifically said that his form of the equations should still be called Maxwell’s equations: Electromagnetic Theory, vol. 1, preface (fifth page), and 69. Hertz agreed: Rautio, “Twenty-three Years,” 1005.

16Heaviside,电磁理论,1:297。

16.Heaviside, Electromagnetic Theory, 1:297.

17吉布斯的向量之路:他首先受到麦克斯韦的启发,然后独立发展——独立于几年后他发现的格拉斯曼。我们从他写给维克多·施莱格尔的信中知道了这一点,这封信后来由吉布斯的学生林德·菲尔普斯·惠勒发表在他的书《约西亚·维拉德·吉布斯:一位伟大思想家的历史》(康涅狄格州纽黑文:耶鲁大学出版社,1952 年)中。

17.Gibbs’s path to vectors: he was first inspired by Maxwell, then branched out on his own—independently of Grassmann, whom he discovered several years later. We know this from his letter to Victor Schlegel, published much later by Gibbs’s student Lynde Phelps Wheeler, in his book Josiah Willard Gibbs: The History of a Great Mind (New Haven, CT: Yale University Press, 1952).

18吉布斯对泰特的“怪物”的回应:《论四元数在矢量代数中的作用》,《自然》第 43 卷(1891 年 4 月 2 日):511-13。海维赛德(包括“雌雄同体怪物”的引文和引文):《电磁理论》,1 卷:137-138 页,第 301 页。

18.Gibbs’s reply to Tait’s “monster”: “On the Role of Quaternions in the Algebra of Vectors,” Nature 43 (April 2, 1891): 511–13. Heaviside (incl. “hermaphrodite monster” quote and citation): Electromagnetic Theory, 1:137–38, 301.

19Peter Guthrie Tait,《四元数在向量代数中的作用》,《自然》第 43 卷(1891 年 4 月 30 日):第 608 页。有关我所引用的向量战争的详细分析,请参阅 Michael J. Crowe 所著的《向量分析史》(印第安纳州圣母大学:圣母大学出版社,1967 年),第 6 章

19.Peter Guthrie Tait, “The Role of Quaternions in the Algebra of Vectors,” Nature 43 (April 30, 1891): 608. For a detailed analysis of the vector wars on which I’ve gratefully drawn, see Michael J. Crowe, A History of Vector Analysis (Notre Dame, IN: University of Notre Dame Press, 1967), chap. 6.

20汤姆森的“四元数之战”:卡吉尔·吉尔斯顿·诺特,《P. G. 泰特的生平和科学工作》(伦敦:剑桥大学出版社,1911 年),185。

20.Thomson’s “war over quaternions”: Cargill Gilston Knott, The Life and Scientific Work of P. G. Tait (London: Cambridge University Press, 1911), 185.

21Knott,《泰特传》,第 185 页;Alexander Macfarlane,《向量代数原理》,《美国科学促进会会刊》第 40 卷(1891 年,1892 年出版):第 65-117 页;Alexander McAulay,《四元数》作为物理研究的实用工具》,《哲学杂志》,第 5 系列,第 33 卷(1892 年 6 月):477–95;Crowe,《向量史》,第 189–97 页。

21.Knott, Life of Tait, 185; Alexander Macfarlane, “Principles of the Algebra of Vectors,” Proceedings of the American Association for the Advancement of Science 40 (1891, published 1892): 65–117; Alexander McAulay, “Quaternions as a Practical Instrument of Physical Research,” Philosophical Magazine, 5th ser., 33 ( June 1892): 477–95; Crowe, History of Vectors, 189–97.

22马丁·里斯 (Martin Rees) 在《如果科学能拯救我们》(剑桥:Polity Press,2022 年)中提出了解决这些问题的解决方案。

22.Martin Rees offers solutions to these problems in If Science Is to Save Us (Cambridge: Polity Press, 2022).

23McAulay 被 Crowe 引用,《矢量史》,195 页。Ida McAulay:Bruce Scott,《McAulay, Alexander》,《澳大利亚传记词典》 ,adb.anu.edu.au

23.McAulay quoted in Crowe, History of Vectors, 195. Ida McAulay: Bruce Scott, “McAulay, Alexander,” Australian Dictionary of Biography, adb.anu.edu.au.

24泰特对麦考利的评论《自然》 49(1893 年 12 月 28 日):193–94。

24.Tait’s review of McAulay: Nature 49 (December 28, 1893): 193–94.

25Heaviside,《向量与四元数》,《自然》(1893 年 4 月 6 日):引用自 Crowe 的《向量史》,第 200 页。水力发电:Scott,“McAulay,Alexander”;Carol Raabus 和 Leon Compton,“塔斯马尼亚水力发电系统的工程壮举”,ABC 电台,2013 年 7 月 29 日。

25.Heaviside, “Vectors versus Quaternions,” Nature (April 6, 1893): quoted in Crowe, History of Vectors, 200. Hydroelectricity: Scott, “McAulay, Alexander”; Carol Raabus and Leon Compton, “The Engineering Feats of Tasmania’s Hydroelectric System,” ABC Radio, July 29, 2013.

26吉布斯,《四元数和向量代数》,《自然》 47(1893 年 3 月 16 日):463-64。

26.Gibbs, “Quaternions and the Algebra of Vectors,” Nature 47 (March 16, 1893): 463–64.

第九章

CHAPTER 9

1 .格蕾丝·奇泽姆 (Grace Chisholm) 论凯莱 (Cayley):I. Grattan-Guinness,《数学联盟:威廉·亨利和格蕾丝·奇泽姆·杨》,《科学年鉴》第 29 卷,第 2 期(1972 年 8 月):117–18。

1.Grace Chisholm on Cayley: I. Grattan-Guinness, “A Mathematical Union: William Henry and Grace Chisholm Young,” Annals of Science 29, no. 2 (August 1972): 117–18.

2纽纳姆学院的历史https://newn.cam.ac.uk/about/history/history-of-newnham/。1920年,女性终于被允许在牛津大学攻读学位,但直到 1948 年,女性才被允许在剑桥大学攻读学位。

2.History of Newnham: https://newn.cam.ac.uk/about/history/history-of-newnham/. Women were finally allowed to take full degrees at Oxford in 1920 but not at Cambridge until 1948.

3 .凯莱和泰特的信件收录于卡吉尔·吉尔斯顿·诺特所著的《PG 泰特的生平和科学著作》(伦敦:剑桥大学出版社,1911 年),第 154–96 页。

3.Cayley and Tait’s letters are in Cargill Gilston Knott, The Life and Scientific Work of P. G. Tait (London: Cambridge University Press, 1911), 154–96.

4 .在撰写本文时,许多机器做出的预测仍需要在实验室中验证,但即便如此,它们也可以为基因组学研究指明方向。

4.At the time of writing, many of these machine-made predictions still need to be verified in the lab, but even so they can point the way for genomics research.

5判别式在平移下的不变性:二次方程ax 2 + bx + c = 0 有解=b±b24一个c/2一个如果我们将x变换为x′ = x + h,则相关的二次方程现在为ax′ 2 + bx′ + c = 0,其解为=b±b24一个c/2一个。判别式相同,b 2 − 4 ac,但解并不相同:对于平移后的方程,我们有=+时长=b±b24一个c/2一个,这意味着=b2一个时长±b24一个c/2一个;但这不等于原始解,=b±b24一个c/2一个

5.Invariance of the discriminant under translations: The quadratic equation ax2 + bx + c = 0 has the solution x=b±b24ac/2a. If we transform x to x′ = x + h, the associated quadratic equation is now ax′2 + bx′ + c = 0, whose solution is x=b±b24ac/2a. The discriminants are the same, b2 − 4ac, but the solutions are not the same: for the translated equation we have x=x+h=b±b24ac/2a, which implies that x=b2ah±b24ac/2a; but this is not equal to the original solution, x=b±b24ac/2a.

6 .Cayley 和 Tait 的引文:Michael J. Crowe,《向量分析史》(印第安纳州圣母大学:圣母大学出版社,1967 年出版),第 212、214 页。Grace Chisholm 论 Cayley:Grattan-Guinness,《数学联盟》,第 117 页。

6.Cayley and Tait quotes: Michael J. Crowe, A History of Vector Analysis (Notre Dame, IN: University of Notre Dame Press, 1967), 212, 214. Grace Chisholm on Cayley: Grattan-Guinness, “A Mathematical Union,” 117.

7Crowe,《矢量史》,214。

7.Crowe, History of Vectors, 214.

8Crowe,《矢量史》,217。

8.Crowe, History of Vectors, 217.

9他在 1901 年 3 月 27 日写给米列娃马里奇的一封信中提到了与求职有关的反犹太主义:《阿尔伯特·爱因斯坦文集》,第 1 卷,约翰·斯塔切尔、戴维·C·卡西迪和罗伯特·舒尔曼编辑(新泽西州普林斯顿:普林斯顿大学出版社,1987 年;英文增刊由安娜·贝克翻译),文件 94,https://einsteinpapers.press.princeton.edu/vol1-trans/182

9.He mentioned anti-Semitism in connection with his job-hunting in a letter to Mileva Marić on March 27, 1901: Collected Papers of Albert Einstein, vol. 1, ed. John Stachel, David C. Cassidy, and Robert Schulmann (Princeton, NJ: Princeton University Press, 1987; English Supplement translated by Anna Beck), document 94, https://einsteinpapers.press.princeton.edu/vol1-trans/182.

10在上一个尾注 ( https://einsteinpapers.press.princeton.edu/vol1-trans/182 )中提到的 1901 年 3 月 27 日的信中,爱因斯坦期待着“我们两人共同将相对运动方面的研究取得圆满成功”的那一天。一些学者认为这证明爱因斯坦和马里奇曾一起研究相对论,尽管上下文和“圆满成功”一词也表明这可能是他们婚姻计划的一个隐喻(由于亲戚的反对、米列娃难以毕业以及爱因斯坦失业)。据我所知,爱因斯坦在现存写给马里奇的信中,唯一一次(以及随后)提到相对论是在 1899 年 9 月 10 日的一封信中(《爱因斯坦文集》,第 1 卷,文件 54),他在信中告诉米列娃,他已经知道了相对于以太的相对运动如何影响光速,并补充道“够了!”(因为她正在备考):https://einsteinpapers.press.princeton.edu/vol1-trans/155;在 1899 年 9 月 28 日的文件 57 中再次提到,其中没有提到“我们的”理论,在 1901 年 12 月 17 日的文件 128 中也是如此。显然,从她的信和爱因斯坦写给她的信来看,她从未就这个问题发表过评论——相反,她专注于与考试相关的话题。如果我们知道他们在一起时发生了什么,而不需要写信就好了!爱因斯坦肯定怀有早期的梦想,希望他们能一起过科学生活——在本卷第 72 号文件(1900 年 8 月 14 日)中,他告诉她,当他不和她在一起时,他缺乏自信和工作乐趣。马里奇当时的信件很少留存下来:那些留存下来的信件都集中在她结婚、通过文凭考试和开始攻读博士学位的梦想上(她的毕业论文是关于热和能量,而不是相对论)。至于她“为爱因斯坦做数学题”,他们在 1900 年的考试结果表明爱因斯坦擅长数学,而她不及格:第 1 卷第 67 号文件,https://einsteinpapers.press.princeton.edu/vol1-trans/163。当然,考试成绩不是一切,但它确实打消了人们对爱因斯坦数学能力的质疑无能。在我看来,我们可以通过研究阻碍她职业生涯的偏见,而不是声称没有明确证据的事情,来更公正地评价马里奇作为女性科学先驱的地位。有关这段关系以及狭义相对论的更多信息,请参阅我的短篇电子书《年轻的爱因斯坦和 E = mc 2的故事》(悉尼:Ligature,2014 年)及其参考文献。有关米列娃·马里奇作为合著者的证据的非常简短的最新概述,请参阅 Ann Finkbeiner,《爱因斯坦第一任妻子的争议遗产》,《自然》 567(2019 年):28-29。

10.In the letter of March 27, 1901, referred to in the previous endnote (https://einsteinpapers.press.princeton.edu/vol1-trans/182), Einstein looks forward to the day when “the two of us together will have brought our work on the relative motion to a victorious conclusion.” This has been cited by some scholars as evidence that Einstein and Marić were working together on relativity, although the context, and the words “victorious conclusion,” also suggest it might be a metaphor for their marriage plans (being held up by disapproval from relatives, Mileva’s struggle to graduate, and Einstein’s lack of employment). The only prior (and subsequent) times Einstein mentions relativity in the extant letters to Marić are, as far as I could find, in a letter of September 10, 1899 (Collected Papers of Albert Einstein, vol. 1, document 54), where he tells Mileva he’s had an idea about how relative motion with respect to the ether affects the velocity of light, adding, “But enough of that!” (because she is studying for her exams): https://einsteinpapers.press.princeton.edu/vol1-trans/155; and again in document 57, September 28, 1899, where there is no mention of “our” theory, and similarly in document 128, December 17, 1901. Evidently, she never responded with comments on the subject, judging from her letters and Einstein’s letters to her—rather, she was focussed on topics relevant to her exams. If only we knew what went on between them when they were together and didn’t need to write letters! Einstein certainly nourished early dreams that they would have a scientific life together—and in document 72 of this volume (August 14, 1900), he tells her that he lacks self-confidence and pleasure in work when he is not with her. Few of Marić’s letters from this time survive: those that do are focussed on her dreams of getting married, passing her diploma exams, and starting her PhD (and her diploma thesis was on heat and energy, not relativity). As for her “doing Einstein’s maths for him,” their exam results in 1900 suggest that Einstein excelled at maths and she failed: document 67 of vol. 1, https://einsteinpapers.press.princeton.edu/vol1-trans/163. Exam results are not everything of course, but it does put to rest claims of Einstein’s mathematical incompetence. In my view we can do more justice to Marić as a female scientific pioneer by examining the prejudice that blighted her career, rather than claiming things for which there is no clear evidence. More information about the relationship, and about the special theory of relativity, is in my short e-book Young Einstein and the Story of E = mc2 (Sydney: Ligature, 2014), and the references therein. For a very brief, updated overview of the evidence for the claim of Mileva Marić as coauthor, see Ann Finkbeiner, “The Debated Legacy of Einstein’s First Wife,” Nature 567 (2019): 28–29.

11麦克斯韦的想法表达在《大英百科全书》第 9 版(1878 年)第 8 卷:568–72 页的“以太”条目中,而更详细的内容则出现在 1879 年 3 月 19 日(即他去世前几个月)写给戴维·佩克·托德的信中。托德意识到了这个想法的重要性,并将其寄给了斯托克斯,斯托克斯将其转达给皇家学会,后者在其会刊上发表了这篇文章:“已故教授 J. 克拉克·麦克斯韦著《关于通过光以太探测太阳系运动的可能方法》”,皇家学会会刊,1880 年 1 月 22 日,第 108–10 页。迈克尔逊研究了这封信,因为他在托德的办公室工作。另请参阅 Robert Shankland 的“迈克尔逊和他的干涉仪”,《今日物理》第 27 卷,第 1 期。 4 (1974):37,DOI:10.1063/1.3128534;Shankland 附上了一张迈克尔逊干涉仪的照片。然而,在报告其结果的论文中,迈克尔逊和莫雷鉴于他们的负面结果,提到卫星方法可能是未来的一种可能的实验,但他们没有引用麦克斯韦;大概他们不知道他的信已经发表了:Albert A. Michelson 和 Edward W. Morley,《论地球和光以太的相对运动》,《美国科学杂志》,系列 3,34,第 203 期(1887 年 11 月):345。

11.Maxwell’s idea was expressed in his entry “Ether” in the 9th edition of Encyclopaedia Britannica (1878), 8:568–72, and in more detail in a letter to David Peck Todd, March 19, 1879, a few months before he died. Todd recognised its importance and sent it to Stokes, who communicated it to the Royal Society, which published it in its proceedings: “‘On a possible method of detecting the motion of the solar system through the luminiferous ether’ by the late Professor J. Clerk Maxwell,” Proceedings of the Royal Society, January 22, 1880, 108–10. Michelson studied this letter, for he worked in Todd’s office. See also Robert Shankland, “Michelson and His Interferometer,” Physics Today 27, no. 4 (1974): 37, DOI: 10.1063/1.3128534; Shankland includes a photo of Michelson’s interferometer. In their paper reporting their results, however, Michelson and Morley mention the satellite method as a possible future experiment in light of their negative result, but they do not cite Maxwell; presumably they didn’t know his letter had been published: Albert A. Michelson and Edward W. Morley, “On the Relative Motion of the Earth and the Luminiferous Ether,” American Journal of Science, ser. 3, 34, no. 203 (November 1887): 345.

12洛伦兹电子理论:麦克斯韦假设电荷是连续分布的,因此在他的方程中电荷密度项为 ρ,电流密度为J;洛伦兹表明,这些密度是电荷(点)分布的近似值或平均值,麦克斯韦方程在这些点处是奇异的,但在其他地方都成立。爱因斯坦论洛伦兹:阿尔伯特·爱因斯坦,《思想与观点》(1954 年;纽约:三河出版社,1982 年),第 73-76 页,以及巴内什·霍夫曼(与海伦·杜卡斯合作),《爱因斯坦》(弗罗格莫尔:帕拉丁出版社,1975 年),第 98 页。

12.Lorentz electron theory: Maxwell had assumed charge was continuously distributed—hence the charge density term ρ and current density J in his equations; Lorentz showed that these densities are an approximation or average of the distribution of charges (points), and that Maxwell’s equations are singular at these points, but hold everywhere else. Einstein on Lorentz: Albert Einstein, Ideas and Opinions (1954; New York: Three Rivers Press, 1982), 73–76, and Banesh Hoffman (with the collaboration of Helen Dukas), Einstein (Frogmore: Paladin, 1975), 98.

13H. Poincaré,“Sur la dynamique de l'elelectron”,Rendiconti del Circolo Matematica di Palermo 21 (1906):18-76。他已于 1905 年 6 月 5 日向科学院提交了关于这篇论文的初步“注释”,并在几年前就相关想法撰写了文章。对于英文版爱因斯坦 1905 年的论文(最初发表在《物理学年鉴》上):A.爱因斯坦,《论运动物体的电动力学》,载于 H.A.洛伦兹等人的《相对论原理》(纽约:多佛,1952 年),第 37-65 页。

13.H. Poincaré, “Sur la dynamique de l’électron,” Rendiconti del Circolo Matematica di Palermo 21 (1906): 18–76. He’d already presented a preliminary “Note” on this paper, to the Académie des Sciences, June 5, 1905, and had written on related ideas several years earlier. For the English version of Einstein’s 1905 paper (originally published in Annalen der Physik): A. Einstein, “On the Electrodynamics of Moving Bodies,” in H. A. Lorentz et al., The Principle of Relativity (New York: Dover, 1952), 37–65.

14早在 1910 年,一直致力于研究爱因斯坦狭义相对论基础洛伦兹群几何的费利克斯·克莱因就声称,“如果真想,可以把‘关于变换群的不变量理论’一词替换为‘关于群的相对论’”(引自 Yvette Kosmann-Schwarzbach [Bertram E. Schwarzbach 译],《诺特定理:20 世纪的不变性和守恒定律》 [纽约:施普林格出版社,2011 年],第 70 页)。2022 年(引自《如果科学拯救我们》 [剑桥:Polity出版社,2022 年],第 93 页),马丁·里斯建议使用“不变性理论”而不是“相对论”这个名称,这样可以避免“与人类语境中的相对主义产生误导性类比”。

14.As early as 1910, Felix Klein, who had been working on the geometry of Lorentz groups that are fundamental in Einstein’s special theory, claimed that one could, “if one really wanted to, replace the term ‘theory of invariants with respect to a group of transformations’ with the term ‘relativity with respect to a group’” (quoted in Yvette Kosmann-Schwarzbach [translated by Bertram E. Schwarzbach], The Noether Theorems: Invariance and Conservation Laws in the Twentieth Century [New York: Springer, 2011], 70). In 2022 (in If Science Is to Save Us [Cambridge: Polity, 2022], 93), Martin Rees suggested the name “theory of invariance” instead of “relativity” would have avoided “misleading analogies with relativism in human contexts.”

15格蕾丝·奇泽姆 (Grace Chisholm) 和更高的维度:她的回忆被转述于 Grattan-Guinness 的《数学联盟》第 128-29 页。

15.Grace Chisholm and higher dimensions: Her recollection is reproduced in Grattan-Guinness, “A Mathematical Union,” 128–29.

16有关这些 4-D 努力及其背景(包括布尔值和超平方类比)的精彩描述,请参阅 Nicholas Mee 的《天体挂毯:艺术和数学的经纬》(牛津:牛津大学出版社,2020 年)。

16.For a beautiful account of these 4-D efforts and their context (including the Booles, and hyper-square analogy), see Nicholas Mee, Celestial Tapestry: The Warp and Weft of Art and Mathematics (Oxford: Oxford University Press, 2020).

17有关 SR 中 (双) 四元数的现代分析,请参阅 Joachim Lambek 的《赞美四元数》,Comptes Rendues 数学报告,加拿大科学院,35,第 4 期(2013 年):121-36;https://www.math.mcgill.ca/barr/lambek/pdffiles/Quater2013.pdf。Hamilton本人也讨论过双四元数(具有复数系数)。

17.For a modern analysis of (bi-)quaternions in SR, see Joachim Lambek, “In Praise of Quaternions,” Comptes Rendues Mathematical Reports, Academy of Sciences, Canada, 35, no. 4 (2013): 121–36; https://www.math.mcgill.ca/barr/lambek/pdffiles/Quater2013.pdf. Hamilton himself discussed biquaternions (which have complex coefficients).

18“懒狗”:引自迈克尔·怀特和约翰·格里宾所著《爱因斯坦:科学人生》(伦敦:西蒙与舒斯特出版社,1993 年),第 39 页。闵可夫斯基论四元数:斯科特·沃尔特,“打破四维矢量:引力中的四维运动,1905-1910”,《广义相对论的起源》,尤尔根·雷恩主编(多德雷赫特:斯普林格出版社,2007 年),第 3 页:212 页。

18.“Lazy dog”: quoted in Michael White and John Gribbin, Einstein: A Life in Science (London: Simon and Schuster, 1993), 39. Minkowski on quaternions: Scott Walter, “Breaking in the 4-vectors: The Four-dimensional Movement in Gravitation, 1905–1910,” in The Genesis of General Relativity, ed. Jürgen Renn (Dordrecht: Springer, 2007), 3:212.

19Minkowski 引用自 Constance Reid 所著的《Hilbert》(柏林:Springer-Verlag,1970 年),105、112。

19.Minkowski quoted in Constance Reid, Hilbert (Berlin: Springer-Verlag, 1970), 105, 112.

20间隔告诉您如何考虑在两个不同地点和时间发生的两个事件之间的“距离”:

20.The interval tells you how to take account of the “distance” between two events taking place at two different places and times:

212+212+212c212

x2x12+y2y12+z2z12ct2t12.

如果你从空间中的同一点连续观察两个事件,那么间隔会根据你自己的手表(因为x 2x 1y 2y 1z 2z 1都为零)。这被称为“本征”时间。类似地,如果你同时测量两个事件,度量会告诉你它们之间的(本征)距离(因为t 2t 1现在为零)。但正如洛伦兹变换所示,相对移动的观察者对这些时间和距离并不一致——你们双方只同意整个间隔是不变的

If you observe two events, one after the other, from the same point in space, then the interval tells you the time between them according to your own wristwatch (because x2x1, y2y1, z2z1 are all zero). This is called the “proper” time. Similarly, if you measure two events at the same time, the metric tells you the (proper) distance between them (because t2t1 is now zero). But as the Lorentz transformations show, there is no agreement from a relatively moving observer on these times and distances—both of you only agree that the interval as a whole is invariant.

顺便说一句,为了将这个二次区间测度的“签名”(正如它的名字)变成 + + + + 而不是 + + + −,闵可夫斯基将时间变为虚数,以符合二次型的原始思想。

By the way, to turn the “signature” (as it’s called) of this quadratic interval measure into + + + + rather than + + + − Minkowski made the time imaginary, to fit with the original idea of a quadratic form.

21H. Minkowski,《空间与时间》,1908 年,英文译本,载于 H.A. 洛伦兹等人所著《相对论原理》,第 75-91 页。红色部分:Reid,Hilbert,第 92 页。关于 1907 年的演讲:Walter,《打破四维向量》,第 219 页。

21.H. Minkowski, “Space and Time,” 1908, English translation in H. A. Lorentz et al., Principle of Relativity, 75–91. Turning red: Reid, Hilbert, 92. On the 1907 lecture: Walter, “Breaking in the 4-vectors,” 219.

22关于闵可夫斯基的逝世:Reid,Hilbert,115。

22.On Minkowski’s death: Reid, Hilbert, 115.

23克莱因和贾斯特斯:克罗,《向量史》,92。

23.Klein and Justus: Crowe, History of Vectors, 92.

24不变的时空间隔/常数 c:速度是距离/时间,因此在三维空间中,光速可以用毕达哥拉斯距离定理来定义:

24.Invariant space-time interval/constant c: Speed is distance/time, so in 3-D space, the speed of light can be defined using Pythagoras’s theorem for the distance:

c2 = x2 + y2 + z2)/ t2

c2 = (x2 + y2 + z2)/t2;

当然,这个等式的另一种写法是

another way of writing this equation is, of course,

x2 + y2 + z2( ct ) 2 = 0

x2 + y2 + z2 − (ct)2 = 0.

左边的表达式在洛伦兹变换下不变,这意味着当坐标 ( x, y, z, t ) 和 ( x′, y′, z′, t′ ) 通过洛伦兹变换关联时,仍然可以得到

The expression on the left is invariant under Lorentz transformations, which means that when the coordinates (x, y, z, t) and (x′, y′, z′, t′) are related via a Lorentz transformation, you still get

x 2 + y 2 + z 2 − ( ct ) 2 = x′ 2 + y′ 2 + z′ 2 − ( ct′ ) 2

x2 + y2 + z2 − (ct)2 = x′2 + y′2 + z′2 − (ct′)2.

这意味着x′ 2 + y′ 2 + z′ 2 − ( ct′ ) 2 = 0,所以 ( x′, y′, z′, t′ ) 框架中的光速也必须为c

Which means that x′ 2 + y′ 2 + z′ 2 − (ct′)2 = 0, too, and so the speed of light in the (x′, y′, z′, t′) frame must also be c.

25William Thomson,“弹性数学理论的要素”,伦敦皇家学会哲学汇刊146 (1856):481-98;奥古斯丁·柯西,“关于实验、平衡条件或固体运动内部运动的方程、弹性或非弹性”,数学练习3 (1828):160-87。

25.William Thomson, “Elements of a Mathematical Theory of Elasticity,” Philosophical Transactions of the Royal Society of London 146 (1856): 481–98; Augustin Cauchy, “Sur les equations qui experiment les conditions d’équilibre ou les lois du movement intérieur d’un corps solide, élastique ou nonélastique,” Exercises de Mathématiques 3 (1828): 160–87.

26麦克斯韦,《电磁论》(牛津:克拉伦登出版社,1873 年),2:278-81。

26.Maxwell, Treatise on Electricity and Magnetism (Oxford: Clarendon Press, 1873), 2:278–81.

27闵可夫斯基将这些特殊的二指标量称为“第二类矢量”,索末菲则称它们为“六矢量”。如今,它们被简称为张量——在这种情况下,是反对称二阶或二阶的张量,其中“秩”或“顺序”指的是其分量上的索引数量。(如果这些张量是通过空间而不是在某一点定义的,那么从技术上讲,它们是张量场。)

27.Minkowski had called these particular two-index quantities “vectors of the second kind,” and Sommerfeld called them “six-vectors.” Today they are simply called tensors—in this case, antisymmetric second-rank or second-order tensors, where the “rank” or “order” refers to the number of indices on its components. (If these tensors are defined through space, rather than at one point, then technically they are tensor fields.)

第十章

CHAPTER 10

1 .爱因斯坦的博士学位:巴内什·霍夫曼,《爱因斯坦》(弗罗格莫尔:帕拉丁,1975 年),55 页。格罗斯曼看到了爱因斯坦的伟大:参见我的《年轻的爱因斯坦》及其参考文献。

1.Einstein’s PhD: Banesh Hoffmann, Einstein (Frogmore: Paladin, 1975), 55. Grossmann seeing Einstein’s greatness: see my Young Einstein and references therein.

2阿尔伯特·爱因斯坦,《观念与观点》(1954 年;纽约:三河出版社,1982 年),第 289 页。

2.Albert Einstein, Ideas and Opinions (1954; New York: Three Rivers Press, 1982), 289.

3 .爱因斯坦致索末菲:朱迪思·古德斯坦,《爱因斯坦的意大利数学家》 (罗德岛州普罗维登斯:美国数学学会,2018 年),第 95 页。

3.Einstein to Sommerfeld: Judith Goodstein, Einstein’s Italian Mathematicians (Providence, RI: American Mathematical Society, 2018), 95.

4 .高斯、测量和最小二乘法:马丁·维米尔和安蒂·拉西利亚,《世界地图:数学大地测量学导论》(英国米尔顿公园:泰勒弗朗西斯出版社,2019 年),第 181 页;弗兰克·里德,《钞票上的数学家:卡尔·弗里德里希·高斯》,《抛物线》第 36 卷,第 2 期(2000 年)。尽管高斯于 1825 年从实地工作中退休,但他一直指导这项调查,直到 1844 年调查完成。

4.Gauss, surveying, and least squares: Martin Vermeer and Antti Rasilia, Map of the World: An Introduction to Mathematical Geodesy (Milton Park, UK: Taylor and Francis, 2019), 181; Frank Reid, “The Mathematician on the Bank Note: Carl Friedrich Gauss,” Parabola 36, no. 2 (2000). Although Gauss retired from fieldwork in 1825, he directed the survey until its completion in 1844.

5实际上,闵可夫斯基和爱因斯坦把这个度量的正负号颠倒过来写成了:

5.Actually, Minkowski and Einstein wrote this metric with the plus and minus signs reversed:

ds 2 = − dx 2dy 2dz 2 + c 2 t 2,或ds 2 = c 2 t 2dx 2dy 2dz 2

ds2 = −dx2dy2dz2 + c2t2, or ds2 = c2t2dx2dy2dz2;

符号的选择被称为“特征”,对于我们的目的而言,关键在于时间微分与空间微分具有相反的符号。

the choice of signs is called the “signature,” and for our purposes the key thing is that the time differential has the opposite sign from the spatial ones.

6 .高斯论证的概要:此处(以及其他地方)的页码指的是高斯 1828 年论文的英文版,即1827 年和 1825 年的《曲面的一般研究》 ,作者是卡尔·弗里德里希·高斯,由詹姆斯·莫尔黑德和亚当·希尔特贝特尔译,古腾堡计划,2011 年(源自 1902 年版,普林斯顿:普林斯顿大学图书馆);https://www.gutenberg.org/files/36856/36856-pdf.pdf

6.Outline of Gauss’s argument: The page numbers here (and elsewhere) refer to the English version of Gauss’s 1828 paper, General Investigations of Curved Surfaces of 1827 and 1825, by Karl Friedrich Gauss, translated by James Morehead and Adam Hiltebeitel, Project Gutenberg, 2011 (from the 1902 edition, Princeton: Princeton University Library); https://www.gutenberg.org/files/36856/36856-pdf.pdf.

我提到,对于二维表面,高斯将x、y、z三个坐标转换为两个新变量的函数,他称之为p、q,因此您可以将坐标变换(我将其设为线性)写为

I mentioned that for his 2-D surface Gauss transformed his three x, y, z coordinates to functions of two new variables, which he called p, q—so you can write the coordinate transformations (which I’ll make linear) as

x = f ( p,q ),y = g ( p,q ),z = h ( p,q )。

x = f(p, q), y = g(p, q), z = h(p, q).

然后链式法则给出

Then the chain rule gives

d=fd+fd=一个d+一个d用高斯的符号表示(第 7 页)。

dx=fpdp+fpdq=adp+adq in Gauss’s notation (p. 7).

(不幸的是,高斯和黎曼一样,使用破折号代替不同的字母。)

(Unfortunately Gauss, like Riemann, used dashes instead of different letters.)

类似地,dy = bdp + b′dq,dz = cdp + c′dq。如果对这些表达式求平方并相加,则得到(参见高斯第 18、20 页):

Similarly, dy = bdp + b′dq, dz = cdp + c′dq. If you square these expressions and add, you get (cf. Gauss pp. 18, 20):

dx 2 + dy 2 + dz 2 = ( a 2 + b 2 + c 2 ) dp 2 + 2( aa′ + bb′ + cc′ ) dpdq +

( a′ 2 + b′ 2 + c′ 2 ) dq 2 = Edp 2 + 2 Fdpdq + Gdq 2 ,

dx2 + dy2 + dz2 = (a2 + b2 + c2)dp2 + 2(aa′ + bb′ + cc′)dpdq +

(a′2 + b′2 + c′2)dq2 = Edp2 + 2Fdpdq + Gdq2,

其中高斯使用E,F,G来简化表达式。

where Gauss used E, F, G to simplify the expression.

但事后看来,这里有一件很有趣的事情:从标量积的代数定义中,你可以看到E、F、G就是我们现在所说的矢量的标量积

But here’s the fascinating thing in hindsight: from the algebraic definition of scalar products, you can see that E, F, G are what we would now call scalar products of the vectors

v = ai + bj + ck v = a′i + b′j + c′k

v = ai + bj + ck, v′ = a′i + b′j + c′k;

换句话说,

in other words,

E = vv,F = vv′,G = v′v′

E = vv, F = vv′, G = v′v′.

这两个向量是坐标线p、q的单位切向量,如叙述中的图 10.4所示。(为了理解这一点,考虑无穷小位移向量,使用向量的简写括号符号:

These two vectors are the unit tangent vectors to the coordinate lines p, q as in fig. 10.4 in the narrative. (To see this, consider the infinitesimal displacement vector, using shorthand bracket notation for vectors:

d r = ( dx , dy , dz ) = ( a , b , c ) dp + ( a′ , b′ , c′ ) dq = v dp + v′ dq ;

dr = (dx, dy, dz) = (a, b, c)dp + (a′, b′, c′)dq = vdp + v′dq;

通过对两个坐标求导,可以找到切向量,就像我们求普通函数的切线斜率一样:

the tangent vectors are found by differentiating this with respect to the two coordinates, just as we differentiate an ordinary function to find the slope of its tangent:

drd=drd=

drdp=v,drdq=v.)

两个向量的标量积的几何定义是ab = | a || b | cosθ,正如我们在第 9 章中看到的那样,|一个|=一个一个。因此这两个切向量之间的角度是

The geometric definition of the scalar product of two vectors is a.b = |a||b| cosθ, and as we saw in chapter 9, |a|=aa. So the angle between these two tangent vectors is

余弦θ==F

cosθ=vv(vv)(vv)=FEG.

后来数学家将其推广到任意度量和维度,其中度量中的系​​数写为g ij:对于这里的二维情况,我们有

Later mathematicians will generalise this to arbitrary metrics and dimensions, where the coefficients in the metric are written as gij: for the 2-D case here we’d have

余弦θ=121122

cosθ=g12g11g22.

为了找到表面的曲率,这些公式被应用于边由坐标线界定的三角形,如图10.4所示,然后,正如我在叙述中所解释的那样,角度的总和给出了曲率的性质。

To find the curvature of the surface, these formulae are applied to triangles whose sides are bounded by the coordinate lines, as in fig. 10.4, and then, as I explained in the narrative, the sum of the angles gives the nature of the curvature.

顺便说一句,如果你熟悉二重积分,那么表达式F2我在叙述中提到的面积积分是雅可比矩阵。它是度量系数矩阵的行列式,并且以广义相对论中的一般度量来表示,它写为

By the way, if you’re familiar with double integrals, then the expression EGF2 in the area integral I mentioned in the narrative is the Jacobian. It’s the determinant of the matrix of coefficients of the metric, and in terms of a general metric, as in GR, it is written as g.

高斯从角度角度对曲率的定义第 46 页(和第 44 页)。

Gauss’s definition of curvature in terms of angles: p. 46 (and 44).

哈里奥特的作品:参见我的《托马斯·哈里奥特:科学人生》(纽约:牛津大学出版社,2019 年),第 160–61 页,以及约翰·斯蒂尔威尔的《数学及其历史》(纽约:Springer-Verlag,1989 年),第 249–50 页。

Harriot’s work: see my Thomas Harriot: A Life in Science (New York: Oxford University Press, 2019), 160–61, and also John Stillwell, Mathematics and Its History (New York: Springer-Verlag, 1989), 249–50.

7霍金关于黑洞视界(边界)的观点:他于 1972 年发表了这一结果;他还在 SW Hawking 和 GFR Ellis 的《时空的大尺度结构》(剑桥:剑桥大学出版社,1973 年),第 335–37 页中证明了这一点。有关霍金证明的简单概述和曲率的简要历史,请参阅 Greg Galloway,“从地球的形状到黑洞的形状:从亚里士多德到霍金及以后”,迈阿密大学数学系,艺术与科学库珀讲座,2017 年 11 月。有关黑洞历史的有趣概述,请参阅诺贝尔奖网站上 2020 年物理学奖的概述。请注意,一些研究人员认为,事件视界望远镜 (EHT) 首次直接拍摄的黑洞图像实际上可能是引力磁单极子而不是黑洞;他们计算出了在未来进行更精确的 EHT 观测时可以区分这两种可能性的参数:M. Ghasemi-Noedi 等人,《研究 M87* 中引力磁单极子的存在》,《欧洲物理学杂志 C 81》,第 939 期(2021 年);https://doi.org/10.1140/epjc/s10052-021-09696-3

7.Hawking on black hole horizon (boundary): He published this result in 1972; he also proved it in S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-time (Cambridge: Cambridge University Press, 1973), 335–37. For a simple sketch of Hawking’s proof and a useful brief history of curvature, see Greg Galloway, “From the Shape of the Earth to the Shape of Black Holes: Aristotle to Hawking and Beyond,” Miami University’s Mathematics Department, Arts and Sciences Cooper Lecture, November 2017. For an interesting outline of the history of black holes, see the overview on the Nobel Prize website for the 2020 physics prize. Note that some researchers have suggested that the Event Horizon Telescope’s (EHT’s) first direct image of a black hole could, in fact, be that of a gravitomagnetic monopole rather than a black hole; they have calculated parameters that would distinguish the two possibilities when future, more accurate EHT observations are made: M. Ghasemi-Noedi et al., “Investigating the Existence of Gravitomagnetic Monopole in M87*,” European Physics Journal C 81, no. 939 (2021); https://doi.org/10.1140/epjc/s10052-021-09696-3.

8仅举一个例子,有关高斯曲率在研究材料起皱方式中所起的作用以及以创新方式使用这些皱纹的可能性的说明,请参阅 Stephen Ornes 的《皱纹的新数学》,《Quanta 》杂志(2022 年 9 月 22 日);https://www.quantamagazine.org/the-new-math-of-wrinkling-patterns-20220922/

8.To take just one example, for an account that includes the role Gaussian curvature is playing in the study of the way materials wrinkle, and the possibilities for using these wrinkles in innovative new ways, see Stephen Ornes, “The New Math of Wrinkling,” Quanta magazine (September 22, 2022); https://www.quantamagazine.org/the-new-math-of-wrinkling-patterns-20220922/.

9刘易斯·坎贝尔和威廉·加内特,《詹姆斯·克拉克·麦克斯韦的生平》(伦敦:麦克米伦,1882 年),第 324–25 页。

9.Lewis Campbell and William Garnett, The Life of James Clerk Maxwell (London: Macmillan, 1882), 324–25.

10爱因斯坦私人讲师:霍夫曼,爱因斯坦,86–87。高斯论黎曼:雷蒙德·弗勒德和罗宾·威尔逊,《伟大的数学家》(伦敦:Arcturus,2011),160。七十七岁的高斯:古德斯坦,《爱因斯坦的意大利数学家》,31。高斯崩溃:史迪威,《数学及其历史》,253–54。

10.Einstein Privatdozent: Hoffmann, Einstein, 86–87. Gauss on Riemann: Raymond Flood and Robin Wilson, The Great Mathematicians (London: Arcturus, 2011), 160. Seventy-seven-year-old Gauss: Goodstein, Einstein’s Italian Mathematicians, 31. Gauss devastated: Stillwell, Mathematics and Its History, 253–54.

11关于黎曼的工作,有一份更优秀、更专业的描述——以及该论文的英文翻译——载于露丝·法威尔 (Ruth Farwell) 和克里斯托弗·尼 (Christopher Knee) 的《缺失的一环:黎曼的‘评注’、微分几何和张量分析》,《数学史》第 17 卷 (1990):223–55 页。

11.An excellent, more technical account of Riemann’s working—and an English translation of the paper—is in Ruth Farwell and Christopher Knee, “The Missing Link: Riemann’s ‘Commentatio,’ Differential Geometry and Tensor Analysis,” Historia Mathematica 17 (1990): 223–55.

12Maxwell,《电磁论》(牛津:克拉伦登,1873 年),1:333(第 280 条)。请注意,Thomson 和 Tait(在其T&T′ 1:515 中)表示弹性系数是用字母对表示的,就像黎曼所做的那样,而不是像麦克斯韦那样使用指数。

12.Maxwell, Treatise on Electricity and Magnetism (Oxford: Clarendon, 1873), 1:333 (art. 280). Note that Thomson and Tait (in their T&T′, 1:515) denote coefficients of elasticity by pairs of letters, as Riemann did, rather than using indices as Maxwell did.

13如果材料是各向异性的,那么电导率就是一个双指数张量,正如黎曼假设的那样。对于各向同性的材料,热量会向各个方向扩散,因此无需担心方向变化,您可以使用标量表示。

13.Conductivity is a two-index tensor if the material is anisotropic, as Riemann assumed. For isotropic materials the heat spreads out in all directions, so there’s no need to worry about variations with direction, and you can use a scalar representation.

14伯恩哈德·黎曼,由威廉·金登·克利福德译为英文,《论几何基础的假设》,《自然》第 8 卷,第 183 期(1873 年):14-17 页,第 184 期,第 36-37 页。

14.Bernhard Riemann, translated into English by William Kingdon Clifford, “On the Hypotheses Which Lie at the Bases of Geometry,” Nature 8, no. 183 (1873): 14–17, and no. 184, 36–37.

15有关黎曼 1854 年和 1861 年论文以及包括克里斯托费尔在内的追随者的工作的详细分析,请参阅 Olivier Darrigol 的《黎曼曲率之谜》,《数学史》第 42 卷(2015 年):第 47-83 页。另请参阅 Farwell 和 Knee 的《缺失的环节》。克里斯托费尔 1869 年论文的英文译本载于Bas Fagginger Auer 的《重温克里斯托费尔》(硕士论文,乌得勒支大学数学研究所,2009 年)第8 章。

15.For detailed analyses of Riemann’s 1854 and 1861 papers, and the work of followers including Christoffel, see Olivier Darrigol, “The Mystery of Riemann’s Curvature,” Historia Mathematica 42 (2015): 47–83. See also Farwell and Knee, “Missing Link.” An English translation of Christoffel’s 1869 paper is given in chap. 8 of Bas Fagginger Auer’s “Christoffel Revisited” (master’s thesis, Mathematical Institute, University of Utrecht, 2009).

16如果选择单位使得c = 1,则系数为 1、1、1、−1。黎曼证明,如果度量具有常数系数,则可以缩放坐标,使得度量中的所有系数都为 1(或 −1,正如闵可夫斯基后来证明的那样)。

16.If units are chosen so that c = 1, the coefficients are 1, 1, 1, −1. Riemann showed that if the metric has constant coefficients, the coordinates can be scaled so that all the coefficients in the metric are 1 (or −1, as Minkowski later showed).

第十一章

CHAPTER 11

1 .对于里奇的传记细节,包括政治背景,我在本章中引用了朱迪思·古德斯坦 (Judith Goodstein)的《爱因斯坦的意大利数学家》(罗德岛普罗维登斯:美国数学学会,2018 年)。

1.For Ricci’s biographical details, including political context, I’ve drawn throughout this chapter on Judith Goodstein, Einstein’s Italian Mathematicians (Providence, RI: American Mathematical Society, 2018).

2古德斯坦,《爱因斯坦的意大利数学家》,2。

2.Goodstein, Einstein’s Italian Mathematicians, 2.

3 .里奇1872 年 11 月 24 日致安东尼奥·曼佐尼的信,摘自古德斯坦的《爱因斯坦的意大利数学家》第 6 页。

3.Ricci to Antonio Manzoni, November 24, 1872, quoted in Goodstein, Einstein’s Italian Mathematicians, 6.

4 .古德斯坦,《爱因斯坦的意大利数学家》,16。

4.Goodstein, Einstein’s Italian Mathematicians, 16.

5里奇和奇泽姆论克莱因:引自古德斯坦的《爱因斯坦的意大利数学家》,第 16、17 页。注意:索菲·科瓦列夫斯基 1874 年在哥廷根获得的博士学位是“非官方的”,就像奇泽姆在剑桥获得的学位一样。

5.Ricci and Chisholm on Klein: quoted in Goodstein, Einstein’s Italian Mathematicians, 16, 17. NB: Sophie Kovalevsky’s Göttingen doctorate in 1874 was “unofficial,” like Chisholm’s Cambridge degree.

6 .古德斯坦(《爱因斯坦的意大利数学家》,27-30)利用里奇和比安卡的信件描绘了一幅他们求爱的温柔画面。

6.Goodstein (Einstein’s Italian Mathematicians, 27–30) uses Ricci and Bianca’s letters to build a tender picture of their courtship.

7里奇 1884 年论文的引言:引用自古德斯坦的《爱因斯坦的意大利数学家》,第 32 页。

7.Ricci’s introduction to his 1884 paper: quoted in Goodstein, Einstein’s Italian Mathematicians, 32.

8WH 和 G. Chisholm Young,《自然》 58,第 1492 期(1898 年 6 月 2 日):99–100。

8.W. H. and G. Chisholm Young, Nature 58, no. 1492 (June 2, 1898): 99–100.

9黎曼张量的指标:粗略地说,由于该张量由二指标度量分量的二阶导数组成,因此其四个指标与哪个度量分量由哪对坐标微分有关。实际情况比这稍微复杂一些,因为黎曼张量是由度量分量导数的和构成的,但这是一般的想法。

9.Indices on the Riemann tensor: Roughly speaking, since this tensor is made up of second derivatives of the two-index metric components, its four indices relate to which metric component is being differentiated by which pair of coordinates. It’s a little more complicated than this because the Riemann tensor is made of sums of derivatives of the metric components, but this is the general idea.

10麦克斯韦的方法是“加法”——将三个滤光片发出的光相加,然后将图像投射到屏幕上。如今,这种方法在幻灯片、电视和数字图像中都有使用。印刷图像使用“减法”方法(麦克斯韦开创了这一方法后发现),其中三种颜色从纸上的颜料反射出来,而不是通过滤光片/像素层传输到屏幕上。减法方法的三原色与麦克斯韦的“对立”——它们是青色、洋红色和黄色。

10.Maxwell’s process is “additive”—you add the light from the three filters and project the image onto a screen. It is used today in slides and in TV and digital images. Printed images use the “subtractive” method (discovered after Maxwell paved the way), where the three colours are reflected from the pigment on the paper rather than transmitted through the filters/layers of pixels to a screen. The three primary colours of the subtractive method are the “opposites” of Maxwell’s—they are cyan, magenta, and yellow.

11关于人工智能训练模型中数据被盗:例如,请参阅 Nick Vincent 和 Hanlin Li 的《ChatGPT 偷走了你的作品。那么你打算怎么做?》,《连线》(2023 年 1 月 28 日),https://www.wired.com/story/chatgpt-generative-artificial-intelligence-regulation/。关于作家的反击,例如,请参阅 Vanessa Thorpe 的《‘ChatGPT 说我不存在’:作家和艺术家如何反击人工智能》,《卫报》 ,2023 年 3 月 19 日。

11.On the theft of data in training models for AI: See, e.g., Nick Vincent and Hanlin Li, “ChatGPT Stole Your Work. So What Are You Going to Do?,” Wired ( January 28, 2023), https://www.wired.com/story/chatgpt-generative-artificial-intelligence-regulation/. For writers fighting back, see, e.g., Vanessa Thorpe, “‘ChatGPT Said I Did Not Exist’: How Writers and Artists Are Fighting Back against AI,” The Guardian, March 19, 2023.

12NLP(包括 LLM)的优点和问题:到本书付印时,很多事情都会发生变化,因为人工智能的发展速度如此之快,但这里有一些最近的参考资料。有关优点的另一个例子,请参阅 Samantha Spengler 的“对于一些自闭症患者来说,ChatGPT 是一条生命线”,Wired,2023 年 5 月 30 日;https://www.wired.com/story/for-some-autistic-people-chatgpt-is-a-lifeline/#。除了训练数据被盗之外,还有很多问题需要解决,比如 ChatGPT 的部分输出不可靠——因此,尽管它有明显的好处,但人们对其在教育领域的作用尚无定论:例如,参见 Hayden Horner 的《ChatGPT:对教育来说是辉煌还是麻烦》,工程技术学院新闻网站,2023 年 3 月 13 日,https://www.eit.edu.au/chatgpt-brilliance-or-a-bother-for-education/。同样,在远程医疗(以及其他许多领域)中,也有优点和缺点:例如,参见 Som Biswas 的《ChatGPT 在公共卫生中的作用》,《生物医学工程年鉴》(2023 年 3 月),在线发表于https://www.researchgate.net/profile/Som-Biswas-2/publication/369269117。复杂的人工智能存在明显的社会问题,从实现监视到制造深度伪造和虚假新闻。我在第 4 章的注释中提到了偏见问题,但也请参阅 Grace Browne 的文章“人工智能沉浸在科技巨头的‘数字殖民主义’中”,Wired UK(2023 年 5 月 25 日);https://www.wired.co.uk/article/abeba-birhane-ai-datasets。关于类似主题,请参阅《麻省理工学院技术评论》的系列文章“人工智能殖民主义” ,https://www.technologyreview.com/supertopic/ai-colonialism-supertopic/,其中还包括被压迫的人民通过使用人工智能以积极的方式进行反击的例子。然后是环境问题,例如,Maanvi Singh,“随着人工智能行业的蓬勃发展,它将对环境造成什么影响?”,《卫报》(2003 年 6 月 9 日)。关于科学和技术的知情公开辩论比以往任何时候都更加重要!

12.Benefits and problems with NLP including LLMs: Much will have changed by the time this book goes to press, such is the pace of AI development, but here are some recent references. For another example of the benefits, see Samantha Spengler, “For Some Autistic People, ChatGPT Is a Lifeline,” Wired, May 30, 2023; https://www.wired.com/story/for-some-autistic-people-chatgpt-is-a-lifeline/#. On the problems, in addition to the theft of training data, much has been written about the unreliability of some of ChatGPT’s output—so although there are clear benefits, the jury is still out on its role in education: see, e.g., Hayden Horner, “ChatGPT: Brilliance or a Bother for Education,” Engineering Institute of Technology’s news website, March 13, 2023, https://www.eit.edu.au/chatgpt-brilliance-or-a-bother-for-education/. Similarly, in telehealth (and much else), there are advantages and disadvantages: see, e.g., Som Biswas, “Role of ChatGPT in Public Health,” Annals of Biomedical Engineering (March 2023), published online at https://www.researchgate.net/profile/Som-Biswas-2/publication/369269117. There are obvious social problems with sophisticated AI, from enabling surveillance to creating deep fakes and fake news. I mentioned the problem of bias in the notes for chap. 4, but see also, e.g., Grace Browne, “AI Is Steeped in Big Tech’s ‘Digital Colonialism,’” Wired UK (May 25, 2023); https://www.wired.co.uk/article/abeba-birhane-ai-datasets. On a similar theme, see the series of articles “AI Colonialism” by MIT Technology Review, https://www.technologyreview.com/supertopic/ai-colonialism-supertopic/, which also includes examples where oppressed peoples are fighting back by using AI in positive ways. Then there are environmental issues, e.g., Maanvi Singh, “As the AI Industry Booms, What Toll Will It Take on the Environment?,” The Guardian (June 9, 2003). More than ever, informed public debate about science and technology is crucial!

13当然,NLP 和 LLM 的内容远不止张量积。就我个人而言(以及关于 NLP 的更多信息),我特别感谢 Qiuyuan Huang、Paul Smolensky、Xiaodong He、Li Deng、Dapeng Wu,“用于深度 NLP 建模的张量积生成网络”,NAACL-HLT 2018 论文集(新奥尔良):1263–73;Lipeng Ahang 等人,“张量空间中的通用语言模型”,AAAI 第 33 届年会(2019 年);以及 Matthew Kramer,“词嵌入”,Medium.com,2021 年 8 月 31 日。

13.There’s much, much more to NLP and LLMs than tensor products, of course. For my account (and for more on NLP), I’m particularly indebted to Qiuyuan Huang, Paul Smolensky, Xiaodong He, Li Deng, Dapeng Wu, “Tensor Product Generation Networks for Deep NLP Modeling,” Proceedings of NAACL-HLT 2018 (New Orleans): 1263–73; Lipeng Ahang et al., “A Generalized Language Model in Tensor Space,” 33rd Annual Conference of the AAAI (2019); and Matthew Kramer, “Word Embeddings,” Medium.com, August 31, 2021.

14“原则上,一台拥有 300 个量子比特的量子计算机瞬间可以执行的计算量比可见宇宙中的原子数量还要多”:Charles Q. Choi,《量子霸权需要多少个量子比特? 》IEEE 新闻,2020 年 5 月 21 日;https://spectrum.ieee.org/qubit-supremacy# :~:text=Superposition%20lets%20one%20qubit%20perform,eight%20calculations%3B%20and%20so%20on 。

14.“In principle, a quantum computer with 300 qubits could perform more calculations in an instant than there are atoms in the visible universe”: Charles Q. Choi, “How Many Qubits Are Needed for Quantum Supremacy?” IEEE News, May 21, 2020; https://spectrum.ieee.org/qubit-supremacy#:~:text=Superposition%20lets%20one%20qubit%20perform,eight%20calculations%3B%20and%20so%20on.

15Enrico Betti : n维空间中的斯托克斯定理:Victor Katz,“从克莱罗到庞加莱的微分形式史”,数学史8(1981):161-88,特别是 175。Betti作为士兵、期刊撰稿人:Goodstein,《爱因斯坦的意大利数学家》,7。Betti和 Ricci 的论文:Goodstein,《爱因斯坦的意大利数学家》,148。

15.Enrico Betti: Stokes’s theorem in n-D: Victor Katz, “The History of Differential Forms from Clairaut to Poincaré,” Historia Mathematica 8 (1981): 161– 88, esp. 175. Betti as soldier, contributor to journal: Goodstein, Einstein’s Italian Mathematicians, 7. Betti and Ricci’s papers: Goodstein, Einstein’s Italian Mathematicians, 148.

16里奇的信,引用自古德斯坦的《爱因斯坦的意大利数学家》,第 9-10 页。

16.Ricci’s letter, quoted in Goodstein, Einstein’s Italian Mathematicians, 9–10.

17里奇为晋升而进行的长期斗争:古德斯坦,《爱因斯坦的意大利数学家》,35–43,59–61。

17.Ricci’s long fight for promotion: Goodstein, Einstein’s Italian Mathematicians, 35–43, 59–61.

18G. Ricci 和 T. Levi-Civita,“微分绝对计算方法及其应用”,《数学年鉴》 54 (1900):125–201;请参阅 128 了解学习新技能的努力和奖励(我的翻译)。

18.G. Ricci and T. Levi-Civita, “Méthodes de calcul différential absolu et leurs applications,” Mathematische Annalen 54 (1900): 125–201; see 128 for effort and reward in learning a new skill (my translation).

19例如,如果按照图 11.1 ,通过两个逆变向量ab的张量(或外)积形成二阶张量T,则将得到变换规则

19.For instance, if, following fig. 11.1, you form a second-order tensor T via the tensor (or outer) product of two contravariant vectors a and b, you’ll have the transformation rule

电视μ一个μb=一个σμ一个σ一个λ一个λ=一个σμ一个λ一个σ一个λ一个σμ一个λ电视σλ

Tμvaμbv=AσμaσAλvaλ=AσμAλvaσaλAσμAλvTσλ.

不要忘记,这里用于张量、变换矩阵系数和索引的字母是任意的,就像代数中的x一样。但它们提供了多么奇妙的灵活性,因为您可以如此轻松地移动变换符号,为任何您喜欢的张量提供规则!

Don’t forget that the letters used for the tensors, transformation matrix coefficients, and indices here are arbitrary, like x in algebra. But what marvelous flexibility they offer, since you can move the transformation symbols around so easily, giving the rule for any kind of tensor you like!

20Unruh 效应:1976 年,加拿大物理学家 William Unruh 在量子理论的帮助下发现,广义相对论预测的温度并不完全是我在图 11.1中所说的与坐标无关的标量。相反,加速观察者测量的时空温度与静止观察者测量的时空温度略有不同。这种“Unruh 效应”尚未被检测到——你需要以接近光速的速度行进才能检测到一度的温度变化。但在 2022 年,由 James Quach 领导的阿德莱德大学团队发明了一种“量子温度计”,可能很快就会证明 Unruh 和广义相对论是正确的。

20.Unruh effect: In 1976 the Canadian physicist William Unruh found, with the help of quantum theory, that the general theory of relativity predicts that temperature is not exactly the coordinate-independent scalar I said it was in fig. 11.1. Rather, an accelerating observer will measure a slightly different space-time temperature than a stationary observer will. This “Unruh effect” hasn’t yet been detected—you’d need to be traveling close to the speed of light to detect one degree of temperature change. But in 2022 a University of Adelaide team led by James Quach invented a “quantum thermometer” that might very soon prove Unruh, and general relativity, right.

21但请注意,这里我们讨论的是单个框架中的矢量——与上面的旋转示例不同,这个总结与框架之间的变换无关。

21.Note, though, that here we are talking about vectors in a single frame— unlike the rotation example above, this summation is not about transformations between frames.

22证明 nD 坐标变换标量积的不变性:使用变换方程中矩阵系数的微分形式最容易看出这一点。例如,第一个二维旋转变换方程是x′ = x cosθ + y sin θ。使用偏导数,该方程的微分形式为

22.Proving invariance of the scalar product for coordinate transformations in n-D: It’s easiest to see this using the differential form of the matrix coefficients in the transformation equations. For example, the first of the 2-D rotation transformation equations is x′ = x cosθ + y sin θ. Using partial derivatives, the differential form of this equation is

d=d+d

dx=xxdx+xydy,

你可以看到=余弦θ,其他衍生物也是如此——所以这些衍生物就是我标记的成分一个σμ在叙述中。对于列和行向量的标量(或内积),您将得到(使用链式法则得到最后一项):

and you can see that xx=cosθ, and so on for the other derivatives—so that these derivatives are just the components that I labeled Aσμ in the narrative. For the scalar (or inner) product of the column and row vector you’d have (using the chain rule to get the last term):

μμ=一个σμ一个μλσλμσλμσλ=λσσλ

uμvμ=AσμAμλuσvλxμxσxλxμuσvλ=xλxσuσvλ.

重复的指标意味着右边是

The repeated indices mean the right-hand side is

λ11λ+λ22λ+...+λnnλ

xλx1u1vλ+xλx2u2vλ+...+xλxnunvλ.

这些导数是关于独立坐标的,所以唯一有意义的导数是λλ=1。这类似于学校的微积分,我们通常只有一个独立变量,比如x,然后dd=1.。因此,上式右侧表达式中 σ 的唯一可能值是 λ。这意味着我们有

These derivatives are with respect to the independent coordinates, so the only derivative that makes sense is xλxλ=1. It’s analogous to school calculus, where we usually have just one independent variable, say x, and then dxdx=1.. So the only possible value for σ on that right-hand side expression above is λ. Which means we have

u μ´ v μ´ = u λ v λ

uμ´ vμ´ = uλvλ.

变换后的坐标系(带破折号)中的表达式与原始坐标中的表达式相同。(对于重复的索引,我使用什么字母并不重要,因为它们只是占位符,告诉您要求和。因此,我可以将 μ 替换为 λ。)

The expression is the same in the transformed coordinate system (with the dashes) as it is in the original coordinates. (It doesn’t matter what letter I use for the repeated indices, because they are just place-holders telling you to sum. So I can swap μ for λ.)

换句话说,标量积在这种坐标变换下是不变的。

In other words, the scalar product is invariant under this change of coordinates.

23ds2的不变性:我们之前看到,逆变和协变张量的坐标变换矩阵是逆的,一个σμ一个μσ(或者使用矩阵分量的导数符号,μσσμ因此逆对将“抵消”,我们将得到

23.Invariance of ds2: We saw earlier that the coordinate transformation matrices for contravariant and covariant tensors are inverses, AσμAμσ (or using derivative notation for the matrix components, xμxσxσxμ). So the inverse pairs will “cancel,” and we’ll have

ds2=μdμd=一个μσ一个λ一个σμ一个λσλdσdλ=σλdσdλ

ds2=gμvdxμdxv=AμσAvλAσμAλvgσλdxσdxλ=gσλdxσdxλ.

距离测量ds 2在每一帧中具有相同的形式和相同的值。(请记住,重要的是索引的模式,而不是字母的选择。)

The distance measure ds2 has the same form and the same value in each frame. (Remember it’s the pattern of the indices that matters, not the choice of letters.)

24Beltrami 论 Ricci 张量:Goodstein,《爱因斯坦的意大利数学家》,49。

24.Beltrami on Ricci’s tensors: Goodstein, Einstein’s Italian Mathematicians, 49.

25Ricci 和 Levi-Civita,“绝对微分计算方法”,128(我的翻译)。

25.Ricci and Levi-Civita, “Méthodes de calcul différential absolu,” 128 (my translation).

第十二章

CHAPTER 12

1 .G. Ricci 和 T. Levi-Civita,“微分绝对计算方法及其应用”,《数学年鉴》 54 (1900):125–201。

1.G. Ricci and T. Levi-Civita, “Méthodes de calcul différential absolu et leurs applications,” Mathematische Annalen 54 (1900): 125–201.

2RH Dicke(《厄特沃什实验》,《科学美国人》 205,第 6 期(1961 年 12 月):84-95)表示,尚不清楚爱因斯坦在早期思考引力时是否知道厄特沃什的结果,但如果实验证明伽利略定律是错误的,爱因斯坦肯定会听说。对于 2022 年的测试:Pierre Touboul 等人,“显微镜任务:等效原理测试的最终结果”,《物理评论快报》 129(2022 年):121102.1-121102.8。

2.R. H. Dicke (“The Eötvös Experiment,” Scientific American 205, no. 6 (December 1961): 84–95), suggests it’s unclear whether or not Einstein knew about Eötvös’s result during his early thinking about gravity, but that Einstein would certainly have heard if the experiment had shown that Galileo’s law was wrong. For the 2022 test: Pierre Touboul et al., “MICROSCOPE Mission: Final Results of the Test of the Equivalence Principle,” Physical Review Letters 129 (2022): 121102.1–121102.8.

3 .乌尔班·勒威耶 (Urbain LeVerrier) 是第一个计算出这一差异的人;根据更新的测量结果,他的方法得出的误差约为 43 角秒。

3.Urbain LeVerrier was the first to calculate this discrepancy; with updated measurements his method gave about 43 arc seconds.

4 .转引自亚伯拉罕·派斯 (Abraham Pais),《主是微妙的》(Subtle Is the Lord)(牛津:牛津大学出版社,1982 年),第 178 页。有些人将“最幸福”翻译为“最幸运”。

4.Quoted in Abraham Pais, Subtle Is the Lord (Oxford: Oxford University Press, 1982), 178. Some translate “happiest” as “most fortunate.”

5由于下落的观察者越接近地球中心,地球引力的强度就越大,因此下落的观察者和匀速加速的观察者之间存在可测量的差异。同样,海洋潮汐也是由于地球的一侧更靠近月球,并感受到更强烈的引力而引起的。

5.Because the strength of Earth’s gravity increases the closer the falling observer gets to the centre of the Earth, there are measurable differences between the falling observer and the uniformly accelerating one. Similarly, ocean tides are caused because one side of the earth is closer to the moon and feels its pull more strongly.

6 .爱因斯坦在布拉格的任命:巴内什·霍夫曼,《爱因斯坦》(Frogmore:Paladin,1975 年),94。1882 年,布拉格大学(即查理大学)由于捷克民族主义和种族争端而分裂为捷克部分和德国部分:https://cuni.cz/UKEN-298.html

6.Einstein’s appointment at Prague: Banesh Hoffmann, Einstein (Frogmore: Paladin, 1975), 94. In 1882, the University of Prague, known as Charles University, had split into a Czech and a German part in the wake of Czech nationalism and ethnic disputes: https://cuni.cz/UKEN-298.html.

7根据牛顿定律,牛顿引力的潜在形式 F=一个=r2一个=r2。在笛卡尔坐标系中,用X、Y、Z表示重力加速度a的分量,注意矢量a与两个物体之间的距离r方向相同。将一个物体m置于原点,则r是第二个物体M的位置矢量,位于点 ( x, y, z )。a水平分量由以下公式得出一个=一个余弦θ=一个r=r3其余部分也类似。区分这些(使用r=2+2+2和链式法则),然后相加,得到++=0.由于加速度与(保守)力成正比,我们可以将其分量写成势能===因此上述方程就变成了拉普拉斯方程,22+22+22=0.如果物质的密度为 ρ 呈连续分布,则泊松方程成立,右边为 4π G ρ。

7.Potential form of Newtonian gravity, from Newton’s laws F=ma=GmMr2a=GMr2. In Cartesian coordinates, designate the components of the gravitational acceleration a by X, Y, Z, and note that the vector a is in the same direction as r, the distance between the two masses. Putting one mass, m, at the origin, then r is the position vector of the second mass, M, which is at the point (x, y, z). The horizontal component of a is found from ai=acosθ=axr=GMxr3, and similarly for the other components. Differentiating these (using r=x2+y2+z2 and the chain rule), and adding, you get Vx+Yy+Zz=0. Since acceleration is proportional to the (conservative) force, we can write its components in terms of a potential V:X=Vx,Y=Vy,Z=Vz, and so the above equation becomes Laplace’s equation, 2Vx2+2Vy2+2Vz2=0. If there is a continuous distribution of matter with density ρ, then Poisson’s equation holds instead, and the right-hand side is 4πGρ.

8电荷和质量密度注意事项:在电磁学中,需要区分电荷密度和点电荷。在引力的情况下,无论是牛顿引力还是爱因斯坦引力,区别在于物质的平均分布(整个太阳系、星云或星系)和点源(如单个恒星或行星)。(牛顿证明球形物体的行为就像它们的所有质量都集中在一个点,即物体的中心。)这意味着方程在某一点是奇异的(即它们不起作用),但它们在这个点源之外没有问题,在那里它们被称为“真空方程”。它们适用于密度为 ρ 的物质的平均分布。有关更多信息,请参阅 Peter Gabriel Bergmann 的《相对论导论》(纽约:Dover,1976 年),第 175-77 页。

8.Charge and mass density caveats: In electromagnetism there is a need to distinguish between charge density and point charges. In the case of gravity, Newtonian and Einsteinian, the distinction is between an average distribution of matter—across the solar system or in a nebula or galaxy—and a point source like a single star or planet. (Newton proved that spherical bodies act as if all their mass is concentrated at a point, the centre of the body.) What this means is that the equations are singular at a point—that is, they don’t work—but they’re fine outside this point source, where they are called “vacuum equations.” And they’re fine for an average distribution of matter with density ρ. For more, see Peter Gabriel Bergmann, Introduction to the Theory of Relativity (New York: Dover, 1976), 175–77.

9爱因斯坦致贝索,引自汉诺克·古特弗伦德(Hanoch Gutfreund)和尤尔根·雷恩(Jürgen Renn)所著《通向相对论之路:爱因斯坦“广义相对论基础”的历史和意义》(新泽西州普林斯顿:普林斯顿大学出版社,2015 年),第 9 页。“严重错误”引自朱迪思·古德斯坦(Judith Goodstein),《爱因斯坦的意大利数学家》(罗德岛普罗维登斯:美国数学学会,2018 年),第 102–3 页。

9.Einstein to Besso, quoted in Hanoch Gutfreund and Jürgen Renn, The Road to Relativity: The History and Meaning of Einstein’s “The Foundation of General Relativity” (Princeton, NJ: Princeton University Press, 2015), 9. “Serious mistakes” quoted in Judith Goodstein, Einstein’s Italian Mathematicians (Providence, RI: American Mathematical Society, 2018), 102–3.

10引自古德斯坦的《爱因斯坦的意大利数学家》,104。我也非常感谢古德斯坦对亚伯拉罕和爱因斯坦关系的总结。

10.Quoted in Goodstein, Einstein’s Italian Mathematicians, 104. I’m also indebted to Goodstein for my summary of Abraham and Einstein’s relationship.

11爱因斯坦的恳求被他的苏黎世联邦理工学院同学兼教授 Louis Kollros 回忆起来;引用自 N. Straumann 所著《爱因斯坦的‘苏黎世笔记本’和他的广义相对论之旅》,《物理学年鉴》(柏林)523,第 6 期(2011 年):第 488-500 页,特别是第 490 页。

11.Einstein’s plea was recollected by his fellow ETH student and professor Louis Kollros; quoted in N. Straumann, “Einstein’s ‘Zürich Notebook’ and His Journey to General Relativity,” Annals of Physics (Berlin) 523, no. 6 (2011): 488–500, esp. 490.

12爱因斯坦在其1916年的论文(《广义相对论的基础》,1916年,英译本载于HA洛伦兹等人所著的《相对论原理》 [纽约:多佛,1952年],113)中这样定义广义相对论原理:“物理定律必须具有这样的性质,即它们适用于任何运动的参考系。”换句话说,这些定律必须对所有观察者(所有参考系)保持相同的形式,也就是说,它们必须以张量形式表示。

12.In his 1916 paper (The Foundation of the General Theory of Relativity, 1916, English translation in H. A. Lorentz et al., The Principle of Relativity [New York: Dover, 1952], 113), Einstein defined the general principle of relativity this way: “The laws of physics must be of such a nature that they apply to systems of reference in any kind of motion.” In other words, these laws must keep the same form for all observers (all frames of reference), and this means they must be expressed in tensor form.

13阿尔伯特·爱因斯坦,《观念与观点》(1954 年;纽约:三河出版社,1982 年),第 309 页。

13.Albert Einstein, Ideas and Opinions (1954; New York: Three Rivers Press, 1982), 309.

14有坐标变换吗?爱因斯坦必须设法理清许多令人困惑的方面。例如,那些不改变点位置的变换怎么办,比如从笛卡尔坐标到极坐标?爱因斯坦意识到了这个问题,这就是他为何对广义协变性的概念苦苦挣扎的原因,我们将会看到。请参阅 John D. Norton,“广义协变性和广义相对论的基础:八十年的争论”,《物理学进展报告》 56(1993):791-858,特别是第 833-34 页。此外,在流形上,这些变换是针对点的,而不是针对整个空间的。有关这种微妙之处的分析,请参阅 Norton,“广义协变性”,第 804 页;以及约翰·厄尔曼和克拉克·格莱摩,“迷失在张量中:爱因斯坦 1912-1916 年与协方差原理的斗争”,科学史与科学哲学研究9(1978):4,251-78,特别是第 254 页。

14.Any coordinate transformations? There were many confusing aspects that Einstein had to try to sort out. For instance, what about transformations that don’t change the location of points, such as from Cartesian to polar coordinates? Einstein realised the problem, which is why he struggled with the idea of general covariance, as we’ll see. See John D. Norton, “General Covariance and the Foundations of General Relativity: Eight Decades of Dispute,” Reports on Progress in Physics 56 (1993): 791–858, esp. 833–34. Further, on a manifold these transformations are found for points, not for the whole space. For an analysis of this subtlety, see Norton, “General Covariance,” 804; and John Earman and Clark Glymour, “Lost in the Tensors: Einstein’s Struggle with Covariance Principles 1912–1916,” Studies in History and Philosophy of Science 9 (1978): 4, 251–78, esp. 254.

15爱因斯坦,《思想与观点》,288。

15.Einstein, Ideas and Opinions, 288.

16我使用了“最小化”这个词,但从技术上讲,我的意思是“极端化”积分,因为该路线在类时间测地线上是最长的(因为时间膨胀而不是空间收缩);但我们在这里不需要担心这个。

16.I used the word “minimising,” but technically I mean “extremising” the integral, for the route is longest on time-like geodesics (because of time dilation as opposed to space contraction); but we don’t need to worry about this here.

17“着火了”:爱因斯坦的回忆,引自斯特劳曼的《爱因斯坦的‘苏黎世笔记本’》,第 490 页。

17.“Caught fire”: Einstein’s recollection, quoted in Straumann, “Einstein’s ‘Zürich Notebook,’” 490.

18F μν是张量吗?是的,因为(参见第 11 章)它像这样变换:

18.Is Fμν a tensor: yes, because (cf. chap. 11) it transforms like this:

Fμ=一个σμ一个λFσλ

Fμv=AσμAλvFσλ,

在闵可夫斯基时空中,变换矩阵A表示洛伦兹变换 (LT)。然而,在时间和空间坐标交织在一起的 LT 下,其分量为空间和时间(例如速度或电场和磁场矢量)的变换并不像我们在第 11 章中看到的那样简单。

where in Minkowski space-time the transformation matrices A represent the Lorentz transformations (LTs). Under LTs, however, where time and space coordinates are intertwined, vectors whose components are functions of space and time (such as velocity or the electric and magnetic field vectors) don’t transform quite so simply as we saw in chapter 11.

19不同的作者喜欢其中一个名称,但公平性得到了恢复,因为在微分几何中有一个对偶张量,用框中的星号表示,所以两个名称都会使用。

19.Different authors prefer one name or the other, but fairness is restored because in differential geometry there’s a dual tensor, denoted with a star as in the box, so both names get used.

20这种对称性在物理上是通过考虑物质的一个元素来保证的,在数学上是通过考虑降低指标时会发生什么来保证的:μν电视σμ=电视σ, 和μν电视σμ=电视σ但是这些等式的左边是相同的,因为g μν = g νμ,这意味着T νσ = T σν

20.This symmetry is warranted physically by considering an element of the matter and mathematically by considering what happens when you lower the indices: gμνTσμ=Tvσ, and gμνTσμ=Tσv. But the left-hand sides of these equations are the same because gμν = gνμ, which means Tνσ = Tσν.

21局部定律:爱因斯坦的质量能量守恒定律是T μν = 0,但这是一条局部守恒定律。整体引力能量守恒定律仍然特别成问题,但即使是引力场的能量密度等局部概念也很难在物理上定义。这是因为T μν = 0 是一个数学类比;我们将在下一章中看到更多内容。

21.Local laws: Einstein’s mass-energy conservation law is T μν = 0, but this is a local conservation law. The notion of global gravitational energy conservation is still especially problematic, but even local concepts such as the energy density of a gravitational field are hard to define physically. That’s because T μν = 0 is a mathematical analogy; we’ll see more in the next chapter.

在地球上,“局部”区域必须足够小,以至于表面没有可测量的曲率——否则,引力的平方反比定律表明引力因地而异,而我在图 12.1中使用的伽利略的恒定重力加速度 32 英尺/秒/秒不再适用。在太阳系中,“局部”区域可以很大,只要来自太阳和行星的引力场大致恒定。在更远的地方,“局部”可以覆盖很大的区域——也许是两颗恒星之间距离的一半。(这些估计来自伯特兰·罗素的杰作《相对论 ABC》,原版于 1925 年,摘录自《世界物理学和数学宝库》,蒂莫西·费里斯主编 [波士顿:小布朗,1991],第 194-202 页。)

On Earth, a “local” region must be small enough that there’s no measurable curvature of the surface—otherwise the inverse-square law of gravity shows that gravitation varies from place to place, and Galileo’s constant gravitational acceleration of 32 feet/sec/sec, which I used in fig. 12.1, no longer applies. In the solar system, a “local” region can be large, as long as the gravitational field from the sun and planets is roughly constant. Further afield, “local” can cover a huge area—perhaps half the distance between two stars. (I owe these estimates to Bertrand Russell’s brilliant The ABC of Relativity, originally published in 1925, excerpted in The World Treasury of Physics and Mathematics, ed. Timothy Ferris [Boston: Little, Brown, 1991], 194–202.)

有关爱因斯坦在能量守恒和协方差方面的斗争的详细讨论,请参阅 Straumann 的《爱因斯坦的‘苏黎世笔记本’》;Earman 和 Glymour 的《迷失在张量中》;以及 Galina Weinstein 的《为什么爱因斯坦在 1912-1913 年拒绝了十一月张量,却在 1915 年 11 月又回到了它?》,《现代物理学历史与哲学研究》第 62 卷(2018 年):第 98-122 页。“十一月张量”就是里奇张量。

For detailed discussion on Einstein’s struggles with energy conservation and covariance, see Straumann, “Einstein’s ‘Zürich Notebook’”; Earman and Glymour, “Lost in the Tensors”; and Galina Weinstein, “Why Did Einstein Reject the November Tensor in 1912–1913, Only to Come Back to It in November 1915?,” Studies in History and Philosophy of Modern Physics 62 (2018): 98–122. The “November tensor” is the Ricci tensor.

有关爱因斯坦试图解释Entwurf缺乏广义协变性的详细讨论,以及它在今天的哲学意义,请参阅约翰·D·诺顿的《洞论证》,斯坦福哲学百科全书,在线,2019 年更新;https://plato.stanford.edu/entries/spacetime-holearg/

For detailed discussion on Einstein’s attempt to explain Entwurf ’s lack of general covariance, and its philosophical significance even today, see John D. Norton, “The Hole Argument,” Stanford Encyclopedia of Philosophy, online, updated 2019; https://plato.stanford.edu/entries/spacetime-holearg/.

22格罗斯曼 (Grossmann) 谈张量问题:引自 Earman 和 Glymour 的《迷失在张量中》,第 258-59 页。爱因斯坦的“沉重心情”:引自 Straumann 的《爱因斯坦的‘苏黎世笔记本’》,第 489 页。

22.Grossmann on trouble with tensors: quoted in Earman and Glymour, “Lost in the Tensors,” 258–59. Einstein’s “heavy heart”: quoted in Straumann, “Einstein’s ‘Zürich Notebook,’” 489.

23爱因斯坦致艾伦费斯特,文献 173,载于《爱因斯坦文集》,第 8 卷,罗伯特·舒尔曼、A、J. 诺克斯、米歇尔·扬森和约瑟夫·伊利编;英文版译者:Ann M. Hentschel(新泽西州普林斯顿:普林斯顿大学出版社,1998 年);感谢出版社和爱因斯坦论文项目,可在线获取,https://einsteinpapers.press.princeton.edu/vol8-trans/195。爱因斯坦致贝索,引自 Goodstein的《爱因斯坦的意大利数学家》,第 105–6 页。另请参阅 Earman 和 Glymour 的《迷失在张量中》,第 264 页及后续页,其中讨论了爱因斯坦的同事为何拒绝Entwurf

23.Einstein to Ehrenfest, document 173, in Collected Papers of Albert Einstein, vol. 8, ed. Robert Schulman, A, J. Knox, Michel Janssen, and Jósef Illy; English translation by Ann M. Hentschel (Princeton, NJ: Princeton University Press, 1998); available online thanks to the Press and the Einstein Papers Project, https://einsteinpapers.press.princeton.edu/vol8-trans/195. Einstein to Besso, quoted in Goodstein, Einstein’s Italian Mathematicians, 105–6. See also Earman and Glymour, “Lost in the Tensors,” 264ff., for discussion of why Einstein’s colleagues rejected Entwurf.

24引自 Earman 和 Glymour 的《迷失在张量中》第 260 页。

24.Quoted in Earman and Glymour, “Lost in the Tensors,” 260.

25斯特恩在 Hanoch Gutfreund 中引用,“奥托·斯特恩——与爱因斯坦在布拉格和苏黎世”,Springer Link,2021 年 6 月 20 日,开放获取,https://link.springer.com/chapter/10.1007/978-3-030-63963-1_6?error=cookies_not_supported&code=bb7fb68a-a41c-4c71-ace0-33a64d5f7756

25.Stern quoted in Hanoch Gutfreund, “Otto Stern—with Einstein in Prague and in Zürich,” Springer Link, June 20, 2021, open access, https://link.springer.com/chapter/10.1007/978-3-030-63963-1_6?error=cookies_not_supported&code=bb7fb68a-a41c-4c71-ace0-33a64d5f7756.

26米列娃的悲伤信件:引自罗杰·海菲尔德和保罗·卡特的《爱因斯坦的私生活》(伦敦:费伯出版社,1993 年),第 128 页。爱因斯坦对米列娃的尖刻要求在 1914 年 7 月的信件中被痛苦地概述,例如文件 22,《爱因斯坦文集》,第 8 卷,https://einstein papers.press.princeton.edu/vol8-trans/60

26.Mileva’s sad letter: quoted in Roger Highfield and Paul Carter, The Private Lives of Albert Einstein (London: Faber and Faber, 1993), 128. Einstein’s acrimonious demands on Mileva are painfully outlined in letters of July 1914, e.g., document 22, Collected Papers of Albert Einstein, vol. 8, https://einstein papers.press.princeton.edu/vol8-trans/60.

27致文化世界的宣言:康斯坦斯·里德,希尔伯特(柏林:Springer-Verlag,1970 年),137–38。

27.Declaration to the Cultural World: Constance Reid, Hilbert (Berlin: Springer-Verlag, 1970), 137–38.

28爱因斯坦致列维-奇维塔,文件 60,《爱因斯坦文集》,第 8 卷,https://einsteinpapers.press.princeton.edu/vol8-trans/99

28.Einstein to Levi-Civita, document 60, Collected Papers of Albert Einstein, vol. 8, https://einsteinpapers.press.princeton.edu/vol8-trans/99.

29David E. Rowe,“爱因斯坦遇见希尔伯特:物理学和数学的十字路口”,《物理学展望》 3(2001):379–424,特别是第 393–96 页。

29.David E. Rowe, “Einstein Meets Hilbert: At the Crossroads of Physics and Mathematics,” Physics in Perspective 3 (2001): 379–424, esp. 393–96.

30正如我们在第 11 章中看到的那样,齐次坐标变换使诸如ab = 0 之类的方程保持不变。同样,爱因斯坦在他 1916 年的概述(《广义相对论的基础》,1916 年,英文译本载于 H.A. 洛伦兹等人的《相对论原理》 [纽约:多佛,1952],第 121 页)中说道:“如果一条自然法则可以用将张量的所有分量相等为零来表达,那么它一般就是协变的。”有关讨论,请参阅 Norton,《广义协变》,第 833–34 页。诺顿指出(834),度量的协变比张量分析的通常变换更具限制性。

30.As we saw in chapter 11, homogeneous coordinate transformations keep equations such as ab = 0 invariant. Similarly Einstein said, in his 1916 overview (The Foundation of the General Theory of Relativity, 1916, English translation in H. A. Lorentz et al., The Principle of Relativity [New York: Dover, 1952], 121), that “if a law of nature is expressed by equating all the components of a tensor to zero, it is generally covariant.” For discussion, see Norton, “General Covariance,” 833–34. Norton notes (834) that covariance of the metric is more restrictive than the usual transformations of tensor analysis.

31爱因斯坦 1915 年 11 月写给家人的信收录在《爱因斯坦文集》第 8 卷(周围都是写给希尔伯特的信!),例如文件 142-43,https://einsteinpapers.press.princeton.edu/vol8-trans/174,文件 150,https://einsteinpapers.press.princeton.edu/vol8-trans/177。1915年 11 月 5 日,马里奇表示愿意让爱因斯坦多见见他们的儿子,文件 135,https://einsteinpapers.press.princeton.edu/vol8-trans/169。爱因斯坦从来没有和他年轻的儿子保持过良好的关系儿子爱德华聪明但非常敏感,后来患上了精神分裂症。爱因斯坦支付了他的医疗费用,但情感负担却落在了马里奇身上。

31.Einstein’s November 1915 letters to his family are in Collected Papers of Albert Einstein, vol. 8 (surrounded by letters to Hilbert!), e.g., documents 142–43, https://einsteinpapers.press.princeton.edu/vol8-trans/174, document 150, https://einsteinpapers.press.princeton.edu/vol8-trans/177. On November 5, 1915, Marić had indicated her own willingness for Einstein to see more of their boys, document 135, https://einsteinpapers.press.princeton.edu/vol8-trans/169. Einstein never managed a good relationship with his younger son, Eduard, who was brilliant but highly sensitive, later developing schizophrenia. Einstein paid for his care, but the emotional burden fell on Marić.

32爱因斯坦写给他的朋友埃伦费斯特的信,引自巴内什·霍夫曼的《爱因斯坦》(Frogmore: Paladin, 1975),125;爱因斯坦写给希尔伯特的信, 1915 年 11 月 18 日, 《爱因斯坦文集》,第 8 卷,文件 148,https://einsteinpapers.press.princeton.edu/vol8-trans/176爱因斯坦写给贝索的信,1915 年 11 月 17 日,《爱因斯坦文集》,第 8 卷,文件 147,https://einsteinpapers.press.princeton.edu/vol8-trans/176。请注意,爱因斯坦还没有完整的场方程,但他有正确的真空方程,这正是他计算水星测地线路径所需要的;结果偏离了牛顿椭圆,这导致他得出了近日点运动的差异。有关推导,请参阅广义相对论教科书,如雷·德因弗诺的《爱因斯坦相对论导论》(牛津:克拉伦登出版社,1992 年),第 195-98 页(其中第 198 页比较了近日点进动的观测值与使用广义相对论计算的值)。

32.Einstein to his friend Ehrenfest, quoted in Banesh Hoffmann, Einstein (Frogmore: Paladin, 1975), 125; Einstein to Hilbert, November 18, 1915, Collected Papers of Albert Einstein, vol. 8, document 148, https://einsteinpapers.press.princeton.edu/vol8-trans/176; Einstein to Besso, November 17, 1915, in Collected Papers of Albert Einstein, vol. 8, document 147, https://einsteinpapers.press.princeton.edu/vol8-trans/176. Note that Einstein did not yet have his full field equations, but he did have the correct vacuum equations, which is what he needed to calculate the geodesic path of Mercury; the result deviated from a Newtonian ellipse, and this gave him the discrepancy in the motion of the perihelion. For the derivation see general relativity textbooks such as Ray d’Inverno, Introducing Einstein’s Relativity (Oxford: Clarendon Press, 1992), 195–98 (including 198 for comparison of observed values of perihelion precession with the values calculated using general relativity).

3311 月 20 日论文中没有明确的方程式:Leo Corry 和他的同事 Jürgen Renn(马克斯·普朗克科学史研究所)和 John Stachel(波士顿大学爱因斯坦研究中心)将证明与“希尔伯特-爱因斯坦优先权之争的迟来决定”中发表的版本进行了比较,《科学》第 278 期(1997 年):1270-73 页。更多参考文献如下。希尔伯特 11 月 20 日的证明通常不协变:参见 Vladimir P. Vizgin,“论爱因斯坦和希尔伯特发现引力场方程式:新材料”,Physics-Uspekhi 44,第 12 期(2001 年):1289 页;Gutfreund 和 Renn,《通向相对论之路》,第 33 页;Jürgen Renn 和 John Stachel,“希尔伯特的物理学基础:从万物理论到广义相对论的组成部分”,《广义相对论的起源》,Jürgen Renn 主编(Dordrecht:Springer,2007 年),第 4 页:858-59 页。

33.No explicit equations in November 20 paper: Leo Corry and his colleagues Jürgen Renn (Max Planck Institute for the History of Science) and John Stachel (Center for Einstein Studies, Boston University) compared the proofs with the published version in “Belated Decision in the Hilbert-Einstein Priority Dispute,” Science 278 (1997): 1270–73. More references are listed below. Hilbert November 20 proofs not generally covariant: see, e.g., Vladimir P. Vizgin, “On the Discovery of the Gravitational Field Equations by Einstein and Hilbert: New Materials,” Physics-Uspekhi 44, no. 12 (2001): 1289; Gutfreund and Renn, Road to Relativity, 33; Jürgen Renn and John Stachel, “Hilbert’s Foundation of Physics: From a Theory of Everything to a Constituent of General Relativity,” in The Genesis of General Relativity, ed. Jürgen Renn (Dordrecht: Springer, 2007), 4:858–59.

34希尔伯特效仿米氏,采用了一种与爱因斯坦截然不同的方法——一种优雅的拉格朗日(变分)方法。但爱因斯坦也已经在 1914 年的论文中尝试过这种方法,希尔伯特读过这篇论文。爱因斯坦也在 1916 年的综述论文中用变分法讨论了守恒定律,使用的是哈密顿量而不是拉格朗日量。有关这两种方法的讨论,请参阅 Rowe,《爱因斯坦遇见希尔伯特》,第 414-15 页。

34.Following Mie, Hilbert used a very different approach from Einstein—an elegant Lagrangian (variational) approach. But Einstein, too, had already tried this approach, in his 1914 paper, which Hilbert had read. Einstein included his variational method for discussing the conservation laws in his 1916 overview paper as well, using a Hamiltonian rather than a Lagrangian. For discussion on the two approaches see Rowe, “Einstein Meets Hilbert,” 414–15.

35爱因斯坦-希尔伯特优先权,以及爱因斯坦和希尔伯特的不同路线:Tilman Sauer,“爱因斯坦方程和希尔伯特作用:希尔伯特关于物理学基础的第一次通讯的证明第 8 页缺少什么?”,精确科学史档案59(2005 年):577-90。Vizgin,“关于引力场方程的发现”,1283-98。Leo Corry、Jürgen Renn 和 John Stachel,“迟来的决定”,1270-73。F. Winterberg,“关于‘希尔伯特-爱因斯坦优先权之争中的迟来的决定’,由 L. Corry、J. Renn 和 J. Stachel 出版”,Z. Naturforsch 59a(2004 年):715-19。 John Earman 和 Clark Glymour,《爱因斯坦和希尔伯特:广义相对论历史中的两个月》,《精确科学史档案》第 19 卷(1978 年):第 291-308 页。Renn 和 Stachel,《希尔伯特的物理学基础》,第 4 卷:第 857-973 页(有关爱因斯坦推导其理论的说明,请参见例如, 《爱因斯坦文集》第 8 卷中他写给索末菲(文件 153)和埃伦费斯特(文件 185)的信)。David E. Rowe,《爱因斯坦遇见希尔伯特》,第 379-424 页。 Ivan T. Todorov,“爱因斯坦和希尔伯特:广义相对论的创造”,预印本在线于 arXiv:physics/0504179v1,2005 年 4 月 25 日。Galina Weinstein,“爱因斯坦是否‘Nostrify’了希尔伯特场方程的最终形式?”,在线于Physics ArXiv:1412.1816,2014年 12 月 11 日。还有更多!请注意,网上有很多关于优先权问题的错误信息。例如,VA Petrov(“爱因斯坦、希尔伯特和万有引力方程”,在线博客https://arxiv.org/pdf/gr-qc/0507136.pdf)声称爱因斯坦不可能推导出水星近日点的正确进动,因为他说他做到了,因为他还没有最终的方程式;他暗示爱因斯坦一定看过希尔伯特的工作(关于迹项),但近日点方程只需要没有迹的真空解;佩特罗夫还暗示希尔伯特推导出了比安奇恒等式,但这掩盖了希尔伯特没有充分理解这些恒等式在能量守恒中的作用的事实(我将在下一章中讨论)。佩特罗夫引用了温特伯格(上文),温特伯格引用了 CJ Bjerknes(佩特罗夫也引用了),但 Bjerknes 并不是一个可信的学者:例如,参见 John Stachel 的《反爱因斯坦情绪再次浮现》,《物理世界》第 16 卷,第 4 期(2003 年):第 40 页。

35.Einstein-Hilbert priority, and on the different routes of Einstein and Hilbert: Tilman Sauer, “Einstein Equations and Hilbert Action: What Is Missing on Page 8 of the Proofs for Hilbert’s First Communication on the Foundations of Physics?,” Archives for History of Exact Sciences 59 (2005): 577–90. Vizgin, “On the Discovery of the Gravitational Field Equations,” 1283–98. Leo Corry, Jürgen Renn, and John Stachel, “Belated Decision,” 1270–73. F. Winterberg, “On ‘Belated Decision in the Hilbert-Einstein Priority Dispute,’ Published by L. Corry, J. Renn and J. Stachel,” Z. Naturforsch 59a (2004): 715–19. John Earman and Clark Glymour, “Einstein and Hilbert: Two Months in the History of General Relativity,” Archives for History of Exact Sciences 19 (1978): 291–308. Renn and Stachel, “Hilbert’s Foundation of Physics,” 4:857–973 (for Einstein’s account of his derivation, see, e.g., his letters to Sommerfeld (document 153) and Ehrenfest (document 185) in Collected Papers of Albert Einstein, vol. 8). David E. Rowe, “Einstein Meets Hilbert,” 379–424. Ivan T. Todorov, “Einstein and Hilbert: The Creation of General Relativity,” preprint online at arXiv:physics/0504179v1, April 25, 2005. Galina Weinstein, “Did Einstein ‘Nostrify’ Hilbert’s Final Form of the Field Equations?,” online at Physics ArXiv:1412.1816, December 11, 2014. And more! Note that there is much misinformation online about the priority issue. For instance, V. A. Petrov (“Einstein, Hilbert and Equations of Gravitation,” blog online at https://arxiv.org/pdf/gr-qc/0507136.pdf ) claims Einstein couldn’t have derived the correct advance of Mercury’s perihelion when he said he did, because he didn’t yet have the final equations; he implies Einstein must have seen Hilbert’s work (on the trace term), but the perihelion equations need only the vacuum solution where there is no trace; Petrov also implies that Hilbert derived the Bianchi identities, but this obscures the fact that Hilbert did not adequately understand the role these identities play in conservation of energy (as I’ll discuss in the next chapter). Petrov cites Winterberg (above), who cites C. J. Bjerknes (as Petrov does), but Bjerknes is not a credible scholar: see, e.g., John Stachel, “Anti-Einstein Sentiment Surfaces Again,” Physics World 16, no. 4 (2003): 40.

36Rowe 在《爱因斯坦遇见希尔伯特》第 408 页中对此进行了很好的讨论;他确实指出(第 418 页),希尔伯特应该更改已发表论文的提交日期,尽管这在当时是标准做法。我应该补充一点,今天发表的论文都附有原始提交日期修订日期。

36.Rowe discusses this well in “Einstein Meets Hilbert,” 408; he does note (418) that Hilbert should have changed the submission date on the published paper, although this was standard practice at the time. I should add that today, papers are published with the original submission date and revision dates.

37Renn 和 Stachel 在《希尔伯特物理学基础》中详细展示了希尔伯特在阅读爱因斯坦的论文后如何修改自己已发表的论文,并指出他最初试图将爱因斯坦的重要贡献据为己有(例如,参见 920–21)。

37.Renn and Stachel, “Hilbert’s Foundation of Physics,” shows in detail the ways that Hilbert modified his published paper after reading Einstein’s and notes the ways he’d originally tried to present essential Einstein contributions as his own (see, e.g., 920–21).

38希尔伯特被引用于 Reid, Hilbert , 142。爱因斯坦那封令人心酸的和解信被引用于 Vizgin 的《关于引力场方程的发现》1289 页。

38.Hilbert quoted in Reid, Hilbert, 142. Einstein’s poignant reconciliation letter is quoted in, e.g., Vizgin, “On the Discovery of the Gravitational Field Equations,” 1289.

39我们之前看到,在闵可夫斯基时空中,标量积(以及散度和)的t分量的符号发生了变化。但这里的重点是,这些都是定义,因此只需关注散度项中的指标模式即可。

39.We saw earlier that in Minkowski space-time the scalar product (and hence the divergence sum) has a sign change for the t-component. But the point here is that these are definitions, so just keep your eye on the pattern of the indices in the divergence terms.

40这是因为总能在某一点找到局部惯性(“自由落体”)框架,在这个框架中狭义相对论成立(协变导数中使用的克里斯托费尔符号为零)。根据张量分析规则,张量方程的形式在任何框架中都是不变的——只要你使用(无扭转)协变而不是偏导数来考虑时空曲率。这条规则还假设我们用一般弯曲度量代替平坦的闵可夫斯基度量。

40.This is because it is always possible to find a locally inertial (“free-falling”) frame at a point, one in which special relativity holds (and the Christoffel symbols used in the covariant derivative are zero). By the rules of tensor analysis, the form of a tensor equation will be invariant in any frame—as long as you use (torsion-free) covariant rather than partial derivatives, to take account of the curvature of space-time. This rule also assumes we are replacing the flat Minkowski metric by the general curved metric.

41然而,在“非相对论”单位中,k = 8πG / c4 其中G是牛顿万有引力定律中的比例常数;这突显了一个事实,爱因斯坦通过类比牛顿方程推导出他的方程式,并确保在地球等弱引力场中,它们简化为牛顿方程。

41.In “nonrelativisitic” units, though, k = 8πG/c4, where G is the proportionality constant in Newton’s law of gravity; this highlights the fact that Einstein derived his equations by analogy with Newton’s and ensured that they reduced to Newton’s in weak gravitational fields such as Earth’s.

42这两个方程是等价的,因为在爱因斯坦的原始方程上收缩指标,并注意到μμ=4(根据我们在第 11 章中看到的定义,g μνg μν为逆),你会发现Rμμ=电视μμR=电视(请注意,在最初的方程中,爱因斯坦实际上以 −k作为系数,但由于这是一个常数,因此我将负号吸收到了k的定义中,以适应今天方程的通常写法。此外,在他的最终方程中,爱因斯坦用克里斯托费尔符号完整地写出了左边 [里奇张量],但他已经将这个表达式定义为R μν。)

42.These two equations are equivalent because contracting the indices on Einstein’s original equation, and noting that gμμ=4 (by definition of gμν and gμν as inverses, as we saw in chap. 11), you find that Rμμ=kTμμR=kT. (Note that in his original equation Einstein actually had −k as his coefficient, but since this is a constant, I’ve absorbed the minus sign into my definition of k, to fit with the way the equation is usually written today. Also, in his final equation Einstein wrote the left-hand side [Ricci tensor] out in full, in terms of Christoffel symbols, but he’d already defined this expression as Rμν.)

43爱因斯坦早期方程R μν = kT μν 与这两个最终形式之间的差异取决于标量T或希尔伯特的等效标量R。(这些标量称为“迹”。)爱因斯坦还是希尔伯特首先意识到这一点是“优先权争议”的核心,因为它只隐含在希尔伯特 11 月 20 日的拉格朗日公式中。由于爱因斯坦和希尔伯特交换了论文,他们肯定相互影响,但他们很可能各自独立迈出了这最后一步,因为他们走的路线不同。今天,希尔伯特的方法被广泛使用,他以与拉格朗日相关的所谓爱因斯坦-希尔伯特作用来纪念。

43.The difference between Einstein’s earlier equation, Rμν = kTμν, and these two final forms hinges on the scalar T or Hilbert’s equivalent scalar R. (These scalars are called “traces.”) Whether Einstein or Hilbert realised this first is at the heart of the “priority dispute,” because it is only implicit in Hilbert’s November 20 Lagrangian formulation. Since Einstein and Hilbert exchanged papers, they certainly influenced each other, but it is likely they each took this final step independently, for they followed different routes. Today, Hilbert’s approach is widely used, and he is commemorated in the so-called Einstein-Hilbert action associated with the Lagrangian.

44爱因斯坦理论的最新测试:参见 Pierre Touboul 等人的《显微镜任务:等效原理测试的最终结果》《物理评论快报》 129,第 21102 期(2022 年 9 月 14 日);Ignazio Ciufolini 等人,“使用 LARES 和 LAGEOS 卫星对广义相对论参考系拖拽效应进行改进测试”,《欧洲物理杂志》C 79,文章编号 872(2019 年);Gemma Conroy,“阿尔伯特·爱因斯坦(再次)是对的:天文学家探测到了来自超大质量黑洞后面的光”,ABC News,2021 年 7 月 29 日,https://www.abc.net.au/news/science/2021-07-29/albert-einstein-astronomers-detect-light-behind-black-hole/100333436; Geraint Lewis,“天文学家看到古老星系因空间膨胀而以慢动作闪烁”(对爱因斯坦关于时间变慢的预测的检验),《对话》,2023 年 7 月 4 日;喷气推进实验室博客(2022 年 8 月 24 日),“NASA 科学家通过测试引力帮助探测暗能量”——他们发现爱因斯坦的方程成立。 (德国波恩大学的 Pavel Kroupa 在《暗物质不存在》中表示不同意,IAE 新闻,2022 年 7 月 12 日。但一项新研究利用广义相对论对引力透镜的预测来绘制暗物质图:Robert Lea,“利用‘宇宙化石’创建的新暗物质图再次证明爱因斯坦是对的”,太空[2023 年 4 月 18 日]。)关于黑洞进动:Brandon Specktor,“宇宙中最极端的黑洞碰撞之一证明了爱因斯坦是对的”,LiveScience,2022 年 10 月 13 日。关于参考系拖拽:例如,请参见 Charles Q. Choi,“时空围绕着一颗死星旋转,再次证明爱因斯坦是对的”,Space.com,2020 年 1 月 31 日。还有更多!

44.Recent tests of Einstein’s theory: See, e.g., Pierre Touboul et al., “MICROSCOPE Mission: Final Results of the Test of the Equivalence Principle,” Physical Review Letters 129, no. 21102 (September 14, 2022); Ignazio Ciufolini et al., “An Improved Test of the General Relativistic Effect of Frame-Dragging Using the LARES and LAGEOS Satellites,” European Physical Journal C 79, article no. 872 (2019); Gemma Conroy, “Albert Einstein Was Right (Again): Astronomers Have Detected Light from Behind a Supermassive Black Hole,” ABC News, July 29, 2021, https://www.abc.net.au/news/science/2021-07-29/albert-einstein-astronomers-detect-light-behind-black-hole/100333436; Geraint Lewis, “Astronomers See Ancient Galaxies Flickering in Slow Motion Due to Expanding Space” (a test of Einstein’s predictions about time slowing down), The Conversation, July 4, 2023; Jet Propulsion Laboratory blog (August 24, 2022), “NASA Scientists Help Probe Dark Energy by Testing Gravity”—they found that Einstein’s equations hold firm. (Pavel Kroupa, University of Bonn, disagrees, in “Dark Matter Doesn’t Exist,” IAE News, July 12, 2022. But a new study used general relativity’s prediction of gravitational lensing to map dark matter: Robert Lea, “New Dark Matter Map Created with ‘Cosmic Fossil’ Shows Einstein Was Right (Again),” Space [April 18, 2023].) On precession of black holes: Brandon Specktor, “One of the Most Extreme Black Hole Collisions in the Universe Just Proved Einstein Right,” LiveScience, October 13, 2022. On frame dragging: See, e.g., Charles Q. Choi, “Spacetime Is Swirling around a Dead Star, Proving Einstein Right Again,” Space.com, January 31, 2020. And much more!

有关日食探险(以及光线在黑洞周围弯曲)的通俗概述,请参阅我的文章“100 年后‘科学革命’”,《宇宙杂志》第 83 期(2019 年):第 29-35 页。请注意,约翰·索尔德纳曾利用牛顿理论预测光线弯曲,得出了广义相对论结果的一半。还请注意,1980 年出现了对探险队领导人存在偏见的指控,但这些指控已被推翻,1919 年的结果得到证实。

For a popular overview of the eclipse expeditions (and light bending around a black hole), see my article “A ‘Revolution in Science’ 100 Years Later,” Cosmos Magazine 83 (2019): 29–35. Note it was Johann Soldner who had earlier used Newton’s theory to predict light bending, getting half the general relativity result. Also note that accusations of bias against the leaders of the expedition arose in 1980, but they have been overturned and the 1919 results confirmed.

有关重力磁学的简要概述,请参阅我的文章“重力电磁学的惊人概念”,《宇宙杂志》第 84 期(2019 年 9 月):第 61-63 页。精简版位于https://cosmosmagazine.com/science/introducing-the-amazing-concept-of-gravito-electromagnetism/

For a brief overview of gravitomagnetism, see my piece “The Amazing Concept of Gravito-electromagnetism,” Cosmos Magazine 84 (September 2019): 61–63. An abridged version is at https://cosmosmagazine.com/science/introducing-the-amazing-concept-of-gravito-electromagnetism/.

我自己的研究是针对主要文章中提到的第二个类比。例如,参见 CBG McIntosh、R. Arianrhod、ST Wade 和 C. Hoenselaers 的《电磁 Weyl 张量:分类和分析》,《古典与量子引力》第 11 卷(1994 年):1555-64 页;以及 R. Arianrhod、A. WC. Lun、CBG McIntosh 和 Z. Perjés 的《磁曲率》,古典与量子引力11 (1994): 2331–35. 爱因斯坦引力常数被添加到一些应用的方程中,特别是暗能量。

My own research has been on the second analogy mentioned in the main article. See, e.g., C. B. G. McIntosh, R. Arianrhod, S. T. Wade, and C. Hoenselaers, “Electric and Magnetic Weyl Tensors: Classification and Analysis,” Classical and Quantum Gravity 11 (1994): 1555–64; and R. Arianrhod, A. W-C. Lun, C. B. G. McIntosh, and Z. Perjés, “Magnetic Curvatures,” Classical and Quantum Gravity 11 (1994): 2331–35. Einstein’s gravitational constant is added to the equations for some applications, especially dark energy.

45爱因斯坦,《思想与观点》,289-90。

45.Einstein, Ideas and Opinions, 289–90.

46爱因斯坦的封面在英文译本中缺失,但最近被艾丽西娅·迪肯斯坦 (Alicia Dickenstein) 找到,她在《关于封面:对数学的隐藏赞美》一文中对爱因斯坦表示了完整的感谢,文章发表在《美国数学学会会刊》ns, 46, no. 1(2009 年 1 月):125–29 页。

46.Einstein’s cover page was missing from English translations but was recently tracked down by Alicia Dickenstein, who gives Einstein’s full acknowledgment in “About the Cover: A Hidden Praise of Mathematics,” Bulletin of the American Mathematical Society, n.s., 46, no. 1 ( January 2009): 125–29.

47Levi-Civita 引用自 Goodstein的《爱因斯坦的意大利数学家》,151,155。

47.Levi-Civita quoted in Goodstein, Einstein’s Italian Mathematicians, 151, 155.

第十三章

CHAPTER 13

1 .克莱因和希尔伯特论诺特小姐:引自伊薇特·科斯曼-施瓦茨巴赫著,伯特伦·E·施瓦茨巴赫译,《诺特定理:20世纪的不变性和守恒定律》(纽约:Springer,2011 年),45,66。

1.Klein and Hilbert on Miss Noether: quoted in Yvette Kosmann-Schwarzbach, translated by Bertram E. Schwarzbach, The Noether Theorems: Invariance and Conservation Laws in the Twentieth Century (New York: Springer, 2011), 45, 66.

2约翰·诺顿在 1993 年的论文《广义协变性和广义相对论的基础》中,不仅对爱因斯坦的奋斗历程进行了精彩的描述,还描述了其他人对他的协变性、相对论和等效性原理的回应或重新解释。他以 20 世纪教科书的演变方式为例,指出仍然存在争议或困惑。当然,正如我在上一章末尾所指出的那样,迄今为止的物理观察已经证实,无论其基础如何,这些方程式都非常有效!

2.In his 1993 paper “General Covariance and the Foundations of General Relativity,” John Norton gave a fascinating account not just of Einstein’s struggles but also of the way others responded to or reinterpreted the meaning of his principles of covariance, relativity, and equivalence. He illustrates this with the way textbook accounts evolved over the twentieth century and points out that there is still controversy or confusion. Of course, as I showed at the end of the previous chapter, physical observations so far have confirmed that the equations work brilliantly, no matter their foundations!

3 .Kosmann-Schwarzbach, Noether Theorems , 3–22给出了 Noether 论文的英文翻译。

3.An English translation of Noether’s paper is given in Kosmann-Schwarzbach, Noether Theorems, 3–22.

4 .在地球这样的弱引力场中,事实证明“动量”的时间分量实际上是粒子的静止质量、引力势能和动能的总和。例如,请参阅 Bernard Schutz 的《广义相对论入门教程》(剑桥:剑桥大学出版社,1985 年),第 190 页。

4.In a weak gravitational field like that on Earth, it turns out that the time component of the “momentum” is, indeed, the sum of the particle’s rest mass, gravitational potential energy, and kinetic energy. See, e.g., Bernard Schutz, A First Course in General Relativity (Cambridge: Cambridge University Press, 1985), 190.

5为了了解诺特定果的本质(她在 1918 年的定理中充分概括了这一结果),让我们回顾一下我们讨论过的那些平移。取a的所有值,从xx + a 的平移形成一个群(如图 9.3中的洛伦兹变换)。与不变性有关的群称为“对称群”。这里的“对称性”是通过V独立于x这一事实来表达的不变性。如果每个方向都适用相同的对称性,那么p的所有分量都守恒,我们有熟悉的动量守恒定律。诺特表明,经典力学狭义相对论的对称群是有限的,但广义相对论的对称群是无限的,因为广义相对论允许所有可能的坐标,因此允许所有可能的点变换。这意味着引力场能量动量的守恒定律确实不同于力学和狭义相对论中的通常守恒定律。有关技术细节,请参阅 Kosmann-Schwarzbach,Noether 定理,56-64。

5.To see the essence of Noether’s result, which she fully generalised in her 1918 theorems, let’s go back to those translations we talked about. Taking all values of a, these translations from x to x + a form a group (like the Lorentz transformations in fig. 9.3). Groups relating to invariance are called “symmetry groups.” The “symmetry” here is the invariance expressed by the fact that V is independent of x. If the same symmetry applies in each direction, then all the components of p are conserved and we have the familiar law of conservation of momentum. What Noether showed was that the symmetry groups for classical mechanics and for special relativity are finite, but the symmetry group in general relativity is infinite, because general relativity allows all possible coordinates and therefore all possible point transformations. This meant that the conservation law for the energy-momentum of the gravitational field is indeed different from the usual conservation laws in mechanics and special relativity. For technical details, see Kosmann-Schwarzbach, Noether Theorems, 56–64.

6 .论物理定律的发散性:Peter Gabriel Bergmann,《相对论导论》(纽约:多佛,1976 年),194-97 页。论诺特定理的发散性:Kosmann-Schwarzbach,《诺特定理》,例如,第 6-10 页。

6.On physical laws as divergence: Peter Gabriel Bergmann, Introduction to the Theory of Relativity (New York: Dover, 1976), 194–97. On divergence in Noether’s theorems: Kosmann-Schwarzbach, Noether Theorems, e.g., 6–10.

7关于广义相对论中能量的定义:罗伯特·M·沃尔德,《广义相对论》(芝加哥:芝加哥大学出版社,1984 年),第 84、286 页,关于诺特定理,第 457 页;SW Hawking 和 GFR Ellis,《时空的大尺度结构》(1973 年;剑桥:剑桥大学出版社,1991 年),第 61-62 页、73-74 页、88-96 页。关于诺特定理:Kosmann-Schwarzbach,《诺特定理》;有关更简单的概述,请参阅 David E. Rowe,《论艾米·诺特定理在相对论革命中的作用》,《数学情报》第 41 卷,第 2 期(2019 年):第 65-72 页。

7.On defining energy in general relativity: Robert M. Wald, General Relativity (Chicago: University of Chicago Press, 1984), 84, 286, and for Noether’s theorem, 457; S. W. Hawking and G. F. R. Ellis, The Large-Scale Structure of Space-time (1973; Cambridge: Cambridge University Press, 1991), 61–62, 73–74, 88–96. On Noether’s theorems: Kosmann-Schwarzbach, Noether Theorems; for a simpler overview, see David E. Rowe, “On Emmy Noether’s Role in the Relativity Revolution,” Mathematical Intelligencer 41, no. 2 (2019): 65–72.

8爱因斯坦坚持认为,真正的广义相对论应该允许任何坐标——他的协方差原理——但我们看到,这允许代表同一点的坐标。这意味着我们可以得到不同形式的度量,但实际上代表的是同一时空——就像我们在笛卡尔和极坐标中看到的圆的方程一样。为了缓解这种情况,四个能量守恒方程(或我们将在下一节中遇到的简化比安奇恒等式)对坐标的选择施加了额外的限制(见注释 10)。

8.Einstein had insisted that any coordinates should be allowed for a truly general theory of relativity—his principle of covariance—but we saw that this allows for coordinates that represent the same point. This means that we can get different forms of the metric that actually represent the same spacetime—just as we saw for the equation of the circle in terms of Cartesian and polar coordinates. To mitigate this situation, the four energy-conservation equations (or the contracted Bianchi identities we’ll meet in the next section) impose additional constraints on the choice of coordinates (see note 10).

9在第 12 章中,T μνT μν 具有相同的内容,因为指标是用闵可夫斯基度量提升的。在广义相对论中,当指标提升时,张量会从度量中获取项,但由于这发生在张量方程的两边,因此方程的基本内容是相同的。

9.In chapter 12, Tμν and Tμν had the same content because the indices were raised with the Minkowski metric. In general relativity, tensors pick up terms from the metric when the indices are raised, but because this happens on both sides of a tensor equation, the essential content of the equation is the same.

10这四个简化的比安奇恒等式为里奇张量提供了额外的信息,这意味着在十个爱因斯坦方程中,只有六个是独立的。这允许自由选择四个坐标(框架),确保每个观察者都推导出相同的物理定律。

10.These four contracted Bianchi identities give additional information about the Ricci tensor, which means that of the ten Einstein equations, only six are independent. This allows the freedom to choose the four coordinates (the frame) arbitrarily, ensuring that every observer deduces the same laws of physics.

11T. Levi-Civita (1917),由 S. Antoci 和 A. Loinger 译,“论爱因斯坦理论中引力张量的解析表达式”,https://arxiv.org/pdf/physics/9906004.pdf关于希尔伯特通过变分方法进行的复杂推导,请参阅David E. Rowe 的第 59 页,“爱因斯坦的引力场方程和比安奇恒等式”,Mathematical Intelligencer 24,第 4 期(2002 年):57–66;Ivan T. Todorov,“爱因斯坦和希尔伯特:广义相对论的创造”,arXiv:physics/0504179v1; David E. Rowe,“Emmy Noether 论广义相对论中的能量守恒”,2019 年 12 月 4 日,预印本在线网址:https://arxiv.org/pdf/1912.03269.pdf,21n32;Carlo Cattani 和 Michelangelo De Maria,“守恒定律和引力波”,《引力的吸引力:广义相对论史的新研究》,John Earman、Michel Janssen 和 John D. Norton 主编(波士顿:Birkhäuser,1993 年),第 67 页。

11.T. Levi-Civita (1917), translated by S. Antoci and A. Loinger, “On the Analystic Expression That Must Be Given to the Gravitational Tensor in Einstein’s Theory,” https://arxiv.org/pdf/physics/9906004.pdf. On Hilbert’s convoluted derivation via a variational approach, see p. 59 of David E. Rowe, “Einstein’s Gravitational Field Equations and the Bianchi Identities,” Mathematical Intelligencer 24, no. 4 (2002): 57–66; Ivan T. Todorov, “Einstein and Hilbert: The Creation of General Relativity,” arXiv:physics/0504179v1; David E. Rowe, “Emmy Noether on Energy Conservation in General Relativity,” December 4, 2019, preprint online at https://arxiv.org/pdf/1912.03269.pdf, 21n32; Carlo Cattani and Michelangelo De Maria, “Conservation Laws and Gravitational Waves,” in The Attraction of Gravitation: New Studies in the History of General Relativity, ed. John Earman, Michel Janssen, and John D. Norton (Boston: Birkhäuser, 1993), 67.

12这意味着标量RT在整个宇宙中都是恒定的。这些标量反映了曲率和质量能量,因此真空中的T应该与物质中的 T 不同。

12.It would mean that the scalars R and T would be constant all throughout the universe. These scalars reflect curvature and mass-energy, so T should be different in a vacuum than amongst matter.

13比安奇恒等式:David E. Rowe,《爱因斯坦引力场方程和比安奇恒等式》,《数学情报》第 24 卷,第 4 期(2002 年):57-66 页;Struik 和 Schouten:Rowe,《爱因斯坦引力场方程》,第 66 页;Kosmann-Schwarzbach,《诺特定理》,第 43 页。

13.Bianchi identities: David E. Rowe, “Einstein’s Gravitational Field Equations and the Bianchi Identities,” Mathematical Intelligencer 24, no. 4 (2002): 57– 66; Struik and Schouten: Rowe, “Einstein’s Gravitational Field Equations,” 66; Kosmann-Schwarzbach, Noether Theorems, 43.

14关于斯特鲁克:在此以及以下段落中,我要感谢 David E. Rowe 的《Dirk Jan Struik 访谈录》,《数学情报》第 11 卷,第 1 期(1989 年):第 14-26 页。访谈还讨论了斯特鲁克的马克思主义思想如何导致他在麦卡锡主义下遭受苦难。斯特鲁克显然很了不起,同样了不起的是,他活到了 106 岁(他于 2000 年去世)。

14.On Struik: Here and in the following paragraphs I’m indebted to David E. Rowe, “Interview with Dirk Jan Struik,” Mathematical Intelligencer 11, no. 1 (1989): 14–26. The interview also discusses how Struik’s Marxist ideas led to his suffering under McCarthyism. Struik was evidently remarkable, and, equally remarkably, he lived to be 106 years old (he died in 2000).

15Struik 在 Rowe 的“Interview with Struik”中引用,19。爱因斯坦在 Constance Reid, Hilbert (Berlin: Springer-Verlag, 1970), 142 中引用。

15.Struik quoted in Rowe, “Interview with Struik,” 19. Einstein quotes in Constance Reid, Hilbert (Berlin: Springer-Verlag, 1970), 142.

16转引自 Reid, Hilbert,143。

16.Quoted in Reid, Hilbert, 143.

17爱因斯坦支持诺特:引用于科斯曼-施瓦茨巴赫,诺特定理,72。魏玛:罗,“诺特在相对论中的作用”,66。

17.Einstein in support of Noether: quoted in Kosmann-Schwarzbach, Noether Theorems, 72. Weimar: Rowe, “Noether’s Role in Relativity,” 66.

18现代代数之母:Norbert Schappacher 和 Cordula Tollmien,《艾米·诺特、赫尔曼·外尔和哥廷根学院:边注》,《数学史》 43(2016):194–97。

18.Mother of modern algebra: Norbert Schappacher and Cordula Tollmien, “Emmy Noether, Hermann Weyl, and the Göttingen Academy: A Marginal Note,” Historia Mathematica 43 (2016): 194–97.

19Struik 引自 Rowe,“Struik 访谈”,16。

19.Struik quoted in Rowe, “Interview with Struik,” 16.

20Struik:Rowe,《对 Struik 的采访》,17。Hodge 引用自 Judith Goodstein 著《爱因斯坦的意大利数学家》(罗德岛州普罗维登斯:美国数学学会,2018 年),165。

20.Struik: Rowe, “Interview with Struik,” 17. Hodge: quoted in Judith Goodstein, Einstein’s Italian Mathematicians (Providence, RI: American Mathematical Society, 2018), 165.

211928 年大会和希尔伯特的激动人心的演讲:Reid,Hilbert,188。

21.1928 congress and Hilbert’s stirring speech: Reid, Hilbert, 188.

22例如,请参阅澳大利亚数学学会公报49,第 1 期(Ole Warnaar 的主席专栏)和第 2 期(Aerwm Pulemotov 的来信)。

22.See, e.g., the Australian Mathematical Society’s Gazette 49, nos. 1 (Ole Warnaar’s President’s column) and 2 (Letters, from Aerwm Pulemotov).

23爱因斯坦和嘉当的引言分别出自埃利·嘉当–阿尔伯特·爱因斯坦:关于绝对平行性的书信 1929–1930,罗伯特·德贝弗主编(新泽西州普林斯顿:普林斯顿大学出版社,1979 年),收录于 1929 年 12 月 8 日和 1930 年 2 月 17 日的书信中。

23.Einstein and Cartan quotes are from the letters of December 8, 1929, and February 17, 1930, respectively, in Elie Cartan–Albert Einstein: Letters on Absolute Parallelism 1929–1930, ed. Robert Debever (Princeton, NJ: Princeton University Press, 1979).

24爱因斯坦致嘉当:载于艾利·嘉当-艾伯特·爱因斯坦,德贝弗主编,第 203 页;爱因斯坦致格罗斯曼夫人:载于巴内什·霍夫曼,《爱因斯坦》(弗罗格莫尔:帕拉丁,1975 年,第 36 页)。

24.Einstein to Cartan: in Elie Cartan-Albert Einstein, ed. Debever, 203; Einstein to Mrs. Grossmann: in Banesh Hoffmann, Einstein (Frogmore: Paladin, 1975, 36.

结语

EPILOGUE

1 .粒子对撞机使物理学家能够建立描述物质及其与各种力相互作用的标准模型,欧洲核子研究中心指出,它不仅有科学效益,还有社会效益:https://home.cern/news/news/cern/society-benefits-investing-particle-physics。然而,前粒子物理学家 Sabine Hossenfelder 对其科学效益持怀疑态度;例如,请参阅她的博客文章https://backreaction.blogspot.com/2022/04/did-w-boson-just-break-standard-model.html 。同样, Quora上的这篇文章表明,LHC 的负面结果对科学至关重要,但这意味着排除流行理论:https://www.quora.com/Was-building-the-Large-Hadron-Collider-worth-it-Did-they-discover-anything-from-it。另请参阅,例如 Nick Scott,“CERN 的宏伟抱负:粒子加速器值得吗?” Varsity(2021 年 1 月 26 日),https://www.varsity.co.uk/science/20486;Tom Hartsfield,“请不要再建造大型强子对撞机”,Big Think(2022 年 6 月 6 日),https://bigthink.com/hard-science/large-hadron-collider-economics/

1.Particle colliders enabled physicists to build the Standard Model describing matter and its interaction with the various forces, and CERN points out that there have also been social benefits as well as scientific ones: https://home.cern/news/news/cern/society-benefits-investing-particle-physics. Former particle physicist Sabine Hossenfelder, however, is one who is skeptical about the scientific benefits; see, e.g., her blog post at https://backreaction.blogspot.com/2022/04/did-w-boson-just-break-standard-model.html. Similarly, this post on Quora suggests negative results in the LHC are vital for science, but in the sense of ruling out popular theories: https://www.quora.com/Was-building-the-Large-Hadron-Collider-worth-it-Did-they-discover-anything-from-it. Also see, e.g., Nick Scott, “CERN’s Grand Ambitions: Are Particle Accelerators Worth It?” Varsity ( January 26, 2021), https://www.varsity.co.uk/science/20486; Tom Hartsfield, “Please, Don’t Build Another Large Hadron Collider,” Big Think ( June 6, 2022), https://bigthink.com/hard-science/large-hadron-collider-economics/.

2实际上,狄拉克使用了E = mc 2平方,这产生了正解和负解:正解是爱因斯坦的普通物质方程;负解E = − mc 2指的是反物质。有关简介,请参阅狄拉克 1933 年诺贝尔奖获奖致辞的摘录,收录于《世界物理学、天文学和数学宝库》(蒂莫西·费里斯主编,波士顿:利特尔,布朗出版社,1993 年),第 80-85 页。有关现代分析,请参阅 Luciano Maiani 和 Omar Benhar 的《相对论量子力学》(佛罗里达州博卡拉顿:CRC Press,2016 年),第 113-16 页。

2.In effect, Dirac used the square of E = mc2, which yielded positive and negative solutions: the positive one is Einstein’s equation for ordinary matter; the negative one, E = −mc2, refers to antimatter. For a brief introduction, see the excerpt from Dirac’s 1933 Nobel Prize address in The World Treasury of Physics, Astronomy and Mathematics, ed. Timothy Ferris (Boston: Little, Brown, 1993), 80–85. For a modern analysis, see Luciano Maiani and Omar Benhar, Relativistic Quantum Mechanics (Boca Raton, FL: CRC Press, 2016), 113–16.

3 .请参阅 Bertha Swirles,《自洽场方法中两个电子的相对论相互作用》,《皇家学会学报 A 157》,第 892 期(1936 年 12 月 2 日):第 680-96 页。

3.See Bertha Swirles, “The Relativistic Interaction of Two Electrons in the Self-Consistent Field Method,” Proceedings of the Royal Society A 157, no. 892 (December 2, 1936): 680–96.

4 .计算数学中的张量,尤其是 NLA:例如,参见 Lek-Heng Lim,“计算中的张量”,Acta Numerica (2021):555–764,DOI:10.1017/S09622 492921000076。

4.Tensors in computational maths, esp. NLA: See, e.g., Lek-Heng Lim, “Tensors in Computation,” Acta Numerica (2021): 555–764, DOI:10.1017/S09622 492921000076.

5爱因斯坦致海因里希·桑格,文献 152,《爱因斯坦文集》https://einsteinpapers.press.princeton.edu/vol8-trans/179。这封信还表明了爱因斯坦对前妻明显阻碍他与儿子汉斯·阿尔伯特和解的怨恨。

5.Einstein to Heinrich Zangger, document 152, Collected Papers of Albert Einstein, https://einsteinpapers.press.princeton.edu/vol8-trans/179. This letter also shows Einstein’s bitterness at his ex-wife for apparently holding up his reconciliation with his son Hans Albert.

指数

INDEX

斜体页码指的是图表。

Page numbers in italics refer to figures.

亚伯拉罕,马克斯,312 ;与爱因斯坦辩论,28081,292,391 n10 ;列维-奇维塔和,294

Abraham, Max, 312; debates with Einstein, 28081, 292, 391n10; Levi-Civita and, 294

抽象概念:代数和,4,10,17,72,80,313 ;历史视角,xiii 向量和的想法,48,61 麦克斯韦和,118,107,110,114,116,163 ;空间72,80;时空206 254,265

abstract concepts: algebra and, 4, 10, 17, 72, 80, 313; historical perspective on, xiii; ideas for vectors and, 48, 61; Maxwell and, 118; quaternions and, 107, 110, 114, 116, 163; space and, 72, 80; space-time and, 206; tensors and, 254, 265

超距作用:电磁学和,13233135138150203365 n14,369 n19,376 n11 麦克斯韦和,132 33135138,369 n19

action-at-a-distance: electromagnetism and, 13233, 135, 138, 150, 203, 365n14, 369n19, 376n11; Maxwell and, 13233, 135, 138, 369n19

Ahmes:π 和的估计,22 – 23,349 n3线积分和,122 时间轴,327

Ahmes: estimates of pi and, 2223, 349n3; line integrals and, 122; timeline for, 327

Airy, George: 拒绝四元数,102

Airy, George: rejects quaternions, 102

代数:抽象概念4,10,17,72,80,313 算法716 阿拉伯人56,9 算术和 6,304,348 n19 天文学1,7 布尔,83,86 – 87 ;积分21 – 23,29 – 41 卡尔达诺和,1116,347 凯莱和,8,81,83,86,8990,160,179,194,332;闭71 交换律和,2,332,356 n2;完成平方和,6,10 – 11,327,346 n9,347 n13,354 n11 复数和 16,348 n20;库仑和,154,174 三次方程​​, 1016,347 n15,348 n18 弯曲空间229,23335,238笛卡尔和,6,9,15 奇森和,345 n9,346 n2 ;爱因斯坦和, 7,9,17,313,316,347 n12 ;电磁学和,3,16,348 n17;欧几里得和,4 5 欧拉348 n20 四维几何和 17 ;广义相对论和,9 几何学和 1、4、6、10-17、346 n9、347 n13、347 nn15-16希腊人 5、7、9、13 ;​​汉密尔顿和, 18 , 14 , 17 , 54 , 57 , 6063 , 6971 , 74 , 76 , 8083 , 86 , 8990 , 99107 , 116 , 151 , 161 , 163 , 17879、183、229、243、33233、345 n1、345 n9、346 n2、358 n9 ;​​ ​Harriot 和,8 9 14 17 29 32 52 61 112 328 348 nn18–19,349 n22,350 n7,353 n5 ;矢量的概念和,43 45 ,52636869;虚数和,2361216347 n10;平方反比定律3436125138154277279315329367 n11,368 n15,393 n21 米和,58,1011,13,328,346 n9法律 70,99,162​63356 n2;解放,1 17 麦克斯韦和,3,7,16 ;美索不达米亚和,xiv 9 11,13 牛顿7,14;诺特和,7,16符号和,812,1517 ;数值线性代数(NLA),323 毕达哥拉斯和,45,346 n5;二次方程和6 7,10,13,15 16,347 n16,348 n19 量子理论14,323,347 n12;四数和,3,78,14,17,76 80,102 – 9,112 16,151,16063,17785,345 n1,346 n4 无线电 16 ;现实,4,17 实数和,2 3;机器人1,3;旋转13,14 和,3 空间和 ,70 94,99 – 100 时空194,196,207,211 符号, 411 , 1417 , 34 , 52 , 6163 , 83 , 99100 , 104 , 151 , 265 , 345 n9, 348 n18, 350 n7,353 n5;对称性和,16Tait346 n3;Tartaglia和,12,347 n1617,242 – 43,246,258 – 65,284 – 86,289,295,300;三维空间14 时间轴,328 29,332 – 33 Wallis14 – 15,348 n18 ;应用题和4,7,911

algebra: abstract concepts and, 4, 10, 17, 72, 80, 313; algorithms and, 716; Arabs and, 56, 9; arithmetic and, 6, 304, 348n19; astronomy and, 1, 7; Boolean, 83, 8687; calculus and, 2123, 2941; Cardano and, 1116, 347; Cayley and, 8, 81, 83, 86, 8990, 160, 179, 194, 332; closure and, 71; commutative law and, 2, 332, 356n2; completing the square and, 6, 1011, 327, 346n9, 347n13, 354n11; complex numbers and, 16, 348n20; Coulomb and, 154, 174; cubic equations and, 1016, 347n15, 348n18; curved space and, 229, 23335, 238; Descartes and, 6, 9, 15; Dodgson and, 345n9, 346n2; Einstein and, 7, 9, 17, 313, 316, 347n12; electromagnetism and, 3, 16, 348n17; Euclid and, 45; Euler and, 348n20; four-dimensional geometry and, 17; general theory of relativity and, 9; geometry and, 1, 4, 6, 1017, 346n9, 347n13, 347nn15–16; Greeks and, 5, 7, 9, 13; Hamilton and, 18, 14, 17, 54, 57, 6063, 6971, 74, 76, 8083, 86, 8990, 99107, 116, 151, 161, 163, 17879, 183, 229, 243, 33233, 345n1, 345n9, 346n2, 358n9; Harriot and, 89, 1417, 2932, 52, 61, 112, 328, 348nn18–19, 349n22, 350n7, 353n5; ideas for vectors and, 43, 45, 5263, 6869; imaginary numbers and, 23, 6, 1216, 347n10; inverse square law and, 34, 36, 125, 138, 154, 277, 279, 315, 329, 367n11, 368n15, 393n21; al-Khwārizmī and, 58, 1011, 13, 328, 346n9; laws and, 70, 99, 16263, 356n2; liberation of, 117; Maxwell and, 3, 7, 16; Mesopotamia and, xiv, 911, 13; Newton and, 7, 14; Noether and, 7, 16; notation and, 812, 1517; numerical linear algebra (NLA), 323; Pythagoras and, 45, 346n5; quadratic equations and, 67, 10, 13, 1516, 347n16, 348n19; quantum theory and, 14, 323, 347n12; quaternions and, 3, 78, 14, 17, 7680, 1029, 11216, 151, 16063, 17785, 345n1, 346n4; radio waves and, 16; reality and, 4, 17; real numbers and, 23; robots and, 1, 3; rotation and, 13, 14; scalar numbers and, 3; space and, 7094, 99100; space-time and, 194, 196, 207, 211; symbolic, 411, 1417, 34, 52, 6163, 83, 99100, 104, 151, 265, 345n9, 348n18, 350n7, 353n5; symmetry and, 16; Tait and, 346n3; Tartaglia and, 12, 347n16; tensors and, 17, 24243, 246, 25865, 28486, 289, 295, 300; three-dimensional space and, 14; timeline for, 32829, 33233; Wallis and, 1415, 348n18; word problems and, 4, 7, 911

代数(Bombelli),328

Algebra (Bombelli), 328

代数微积分:du Châtelet 和,41 几何和 21,74 ;牛顿41,74;兴起2123,29 41,74,324 Wallis 32

algebraic calculus: du Châtelet and, 41; geometry and, 21, 74; Newton and, 41, 74; rise of, 2123, 2941, 74, 324; Wallis and, 32

算法:代数和,716;Alpha-Fold,19192;微积分和,212227313441350 n7 ;卡尔达诺和,111656328347 n16 ;高斯消元法,359 n16 ;向量思想和,56;花剌子密和, 7;牛顿和,323;PageRank,8788336360 n21;四元数和,108186;空间和,82838789;时空和, 19192量和,263;时间轴, 32729,336

algorithms: algebra and, 716; Alpha-Fold, 19192; calculus and, 2122, 27, 31, 34, 41, 350n7; Cardano and, 1116, 56, 328, 347n16; Gaussian elimination, 359n16; ideas for vectors and, 56; al-Khwārizmī and, 7; Newton and, 323; PageRank, 8788, 336, 360n21; quaternions and, 108, 186; space and, 8283, 8789; space-time and, 19192; tensors and, 263; timeline on, 32729, 336

Al-Jabr wa'l muqābalah (al-Khwārizmī), 56

Al-Jabr wa’l muqābalah (al-Khwārizmī), 56

天文学大成》(托勒密),xvixviii5,327

Almagest (Ptolemy), xvixviii, 5, 327

AlphaFold 191-92

AlphaFold, 19192

美国科学促进会,181

American Association for the Advancement of Science, 181

安德烈-玛丽安培:电磁学和, 11011 , 128 , 131 , 138 , 174 , 330 , 365 n14, 369 n21, 371 n23

Ampère, André-Marie: electromagnetism and, 11011, 128, 131, 138, 174, 330, 365n14, 369n21, 371n23

分析力学(拉格朗日)1045,329

Analytical Mechanics (Lagrange), 1045, 329

英国国教徒,64 , 72 - 73 , 81 , 116 , 119 , 160

Anglicans, 64, 7273, 81, 116, 119, 160

动量,9496,33536,362 n30​

angular momentum, 9496, 33536, 362n30

《物理学年鉴》(期刊),281

Annalen der Physik (journal), 281

Annali di Matematica pura e applicata(期刊), 25657

Annali di Matematica pura e applicata (journal), 25657

恩斯特·阿佩尔特:Ausdehnungslehre和,10910

Apelt, Ernst: Ausdehnungslehre and, 10910

阿波罗11号93年

Apollo 11, 93

佩尔加的阿波罗尼乌斯,xix

Apollonius of Perga, xix

阿拉伯数学xxvi56,9,46,64,328

Arabic mathematics, xxvi, 56, 9, 46, 64, 328

阿基米德:微积分和,22 – 27,30 – 31 弯曲空间和,225线积分和,122;穷举法和,2226;时间轴,327

Archimedes: calculus and, 2227, 3031; curved space and, 225; line integrals and, 122; method of exhaustion and, 2226; timeline on, 327

·罗伯特·阿甘58、60、69、330

Argand, Jean Robert, 58, 60, 69, 330

阿里乌斯教,72

Arianism, 72

阿里斯塔古,327

Aristarchus, 327

算术:代数和,6,304,348 n19 积分和,3234;向量和的想法,6064;四元数和,104 空间和,71 – 72,76,356 n2 ;张量 255

arithmetic: algebra and, 6, 304, 348n19; calculus and, 3234; ideas for vectors and, 6064; quaternions and, 104; space and, 7172, 76, 356n2; tensors and, 255

无限算术(沃利斯),3234

Arithmetica Infinitorum (Wallis), 3234

阿姆斯特朗,尼尔,3岁

Armstrong, Neil, 3

数组。请参阅矩阵

arrays. See matrices

Ars Magna伟大的艺术(卡尔达诺)11,328,347 n16

Ars Magna (The Great Art) (Cardano), 11, 328, 347n16

人工智能(AI):ChatGPT,249;数据存储和,88 – 89;道德问题和,88 – 89,360 n22,387 n11,38788 n12 向量的想法44 法学硕士24950,387 n12;机器学习 82,86 90,221,24748,32223;神经网络和,191 92 NLP 和,249 51,387 n12;复杂性,ix 387 n12和,249,387 n12;时间轴337

artificial intelligence (AI): ChatGPT, 249; data storage and, 8889; ethical issues and, 8889, 360n22, 387n11, 38788n12; ideas for vectors and, 44; LLMs and, 24950, 387n12; machine learning and, 82, 8690, 221, 24748, 32223; neural networks and, 19192; NLP and, 24951, 387n12; sophistication of, ix, 387n12; tensors and, 249, 387n12; timeline on, 337

Artis Analyticae Praxis (Harriot), 8 , 32 , 52 , 328 , 347 n10, 349 n22, 350 n8

Artis Analyticae Praxis (Harriot), 8, 32, 52, 328, 347n10, 349n22, 350n8

《孙子兵法》48-49 节,第353

art of war, 4849, 353n5

天文学:艾里和102代数和,1,7 黑洞和,228,302,313,315,337,385 n7,397 n44,401 n12 农和,361 n26 ;星座 xvii – xviii弯曲空间22829 日食98,199,294,3023,317,320,335,397 n44 爱因斯坦和,200,293,322,336;吉尔和,145 希腊语xvxvii,344 n6 汉密尔顿和,1,7,62,65,73,76,102 矢量的想法和,60,62,65-66 ;土著,xvii - xviii,344 n5;国际天文学联合会,xviii 337 麦克斯韦,145 托勒密 xviii ; 102,114 红移和 278-81,336-37 萨默维尔和,7 空间和,73,76 时空200;星星和76,280,289,294,303,32122,337,361 n26,391 n8,393 n19,393 n21,397 n44 和,276,293时间轴开启 328,330,336 37​​​​

astronomy: Airy and, 102; algebra and, 1, 7; black holes and, 228, 302, 313, 315, 337, 385n7, 397n44, 401n12; Cannon and, 361n26; constellations, xviixviii; curved space and, 22829; eclipses, 98, 199, 294, 3023, 317, 320, 335, 397n44; Einstein and, 200, 293, 322, 336; Gill and, 145; Greek, xvxvii, 344n6; Hamilton and, 1, 7, 62, 65, 73, 76, 102; ideas for vectors and, 60, 62, 6566; indigenous, xviixviii, 344n5; International Astronomical Union, xviii, 337; Maxwell and, 145; Ptolemy and, xviii; quaternions and, 102, 114; redshift and, 27881, 33637; Somerville and, 7; space and, 73, 76; space-time and, 200; stars and, 76, 280, 289, 294, 303, 32122, 337, 361n26, 391n8, 393n19, 393n21, 397n44; tensors and, 276, 293; timeline on, 328, 330, 33637

Ausdehnungslehre(格拉斯曼),10311 0, 331;汉密尔顿和,112-14 四元数和, 10316 , 163 , 178 ;张量和,256

Ausdehnungslehre (Grassmann), 103110, 331; Hamilton and, 11214; quaternions and, 10316, 163, 178; tensors and, 256

奥斯汀,简,66 , 73 , 145

Austen, Jane, 66, 73, 145

公理163,220,295,333​​

axioms, 163, 220, 295, 333

查尔斯·巴贝奇,62 , 74 , 149 , 358 n8

Babbage, Charles, 62, 74, 149, 358n8

弹道学,12,47-49

ballistics, 12, 4749

海因里希·巴尔泽:关于Ausdehnungslehre110

Baltzer, Heinrich: on Ausdehnungslehre, 110

巴里,JM:Tait 街145 号

Barrie, J. M.: on Tait, 145

巴罗,艾萨克,27岁

Barrow, Isaac, 27

基础(向量向量空间),77,143,153,171,187,251,265,272

basis (vectors, vector spaces), 77, 143, 153, 171, 187, 251, 265, 272

贝拉维蒂斯,朱斯托,102

Bellavitis, Giusto, 102

贝尔特拉米,尤金尼奥,273

Beltrami, Eugenio, 273

伯努利,雅各布,354 n12

Bernoulli, Jakob, 354n12

伯努利约翰,41,54,105,229,329

Bernoulli, Johann, 41, 54, 105, 229, 329

伯尔尼大学:爱因斯坦的第一份学术工作,232

Bern University: Einstein’s first academic job, 232

Besso, Michele, 281 ; 与爱因斯坦的近日点计算以及29293 , 296

Besso, Michele, 281; perihelion calculations with Einstein and, 29293, 296

贝蒂,恩里科,256

Betti, Enrico, 256

安奇恒等式:希尔伯特和,310395 n35;诺特和,31011336401 n13;张量和3101133536372 n9,395 n35,400 n8,400 n10,401 n13

Bianchi identities: Hilbert and, 310, 395n35; Noether and, 31011, 336, 401n13; tensors and, 31011, 33536, 372n9, 395n35, 400n8, 400n10, 401n13

北斗七星,xvii

Big Dipper, xvii

二进制数字(“位”),xxiv251

binary digits (“bits”), xxiv, 251

二项式表达式,60

binomial expressions, 60

黑洞:爱因斯坦和313 14 引力和,302,31314霍金228,385 n7;光和,228,302,313,315,337,385 n7,397 n44,401 n12 合并 323 彭罗斯和,228,337

black holes: Einstein and, 31314; gravity and, 302, 31314; Hawking and, 228, 385n7; light and, 228, 302, 313, 315, 337, 385n7, 397n44, 401n12; merging of, 323; Penrose and, 228, 337

布莱克伍德,约翰:艾略特,麦克斯韦,泰特和, 16566

Blackwood, John: Eliot, Maxwell, Tait and, 16566

波尔,尼尔斯,95

Bohr, Niels, 95

亚诺斯·博莱艾11516,220

Bolyai, Janos, 11516, 220

拉斐尔·邦贝利1415,328

Bombelli, Rafael, 1415, 328

博内,皮埃尔·奥西安,228

Bonnet, Pierre Ossian, 228

布尔,乔治:代数,83,86-87 四元83,100,206时间轴331

Boole, George: algebra of, 83, 8687; quaternions and, 83, 100, 206; timeline on, 331

博恩,马克斯,321

Born, Max, 321

色子,20,99,320,362n30

bosons, 20, 99, 320, 362n30

布曼,凯蒂,313

Bouman, Katie, 313

罗摩笈多,82,328

Brahmagupta, 82, 328

布林,谢尔盖,87,336

Brin, Sergey, 87, 336

英国科学促进会:克利福德与麦克斯韦相遇于160 年;麦克斯韦与汤姆森相遇于123 年;“科学家”一词的引入于101 年

British Association for the Advancement of Science: Clifford meets Maxwell at, 160; Maxwell meets Thomson at, 123; the term “scientist” introduced at, 101

布鲁姆涂鸦2、42、325、331 汉密尔顿和,1 – 3、18、4142、75、78、97、145、243、325、331 元数和,97、145、243、325、331

Broome Bridge: graffiti of, 2, 42, 325, 331; Hamilton and, 13, 18, 4142, 75, 78, 97, 145, 243, 325, 331; quaternions and, 97, 145, 243, 325, 331

布朗运动,347 n12

Brownian motion, 347n12

拜伦女士,73岁

Byron, Lady, 73

积分代数,21 23,29 41,74,324 算法和,21 22,27,31,34,41,350 n7阿基米德 22 – 27,3031;算术和,32 34,350 n7 到达,18 41 笛卡尔坐标32;链式法则 39,267,383 n6,389 n22,391 n7;收敛31,15355,161,172;旋,152 参见;导数和,2030另请参阅导数);微分,1821另请参阅微积分);散度和,298另请参阅散度);杜夏特莱和,383941;爱因斯坦和,193536;欧几里得和,27;伽利略和,27;几何和,21222631343941350 nn7–8,352 n15;梯度和,152另请参阅梯度);引力和,20 27 34 40 希腊人和,2229;汉密尔顿和,18204165; Harriot 和,27293132350 nn7–8;虚数和,21;积分,20273135 (另积分平方反比定律和,34,36 ;运动定律 34 – 35 莱布尼茨2730,35 – 41,349 n5,350 n6 麦克斯韦,18,35,349 nn1–2 ;穷尽2226;运动和 27,30,34 – 36,39,41 牛顿1820,2641,323,350 nn6–7,351 nn11–13,352 n15 符号,29,32,39 – 40 行星运动和,34 36 ;原理以及, 3441 , 350 n7, 351 nn12–13, 352 n15;毕达哥拉斯定理 和, 22 , 23 ;斜率, 40 , 152 , 383 n6;符号, 39 , 54 , 83 ;切线, 36 , 222 , 22526 , 31516 , 383 n6;张量,17另见张量微积分);速度和, 32 , 41 , 352 n15;沃利斯27、31-34、350 nn7-8 ​芝诺和, 2631 , 40 , 44

calculus: algebraic, 2123, 2941, 74, 324; algorithms and, 2122, 27, 31, 34, 41, 350n7; Archimedes and, 2227, 3031; arithmetic and, 3234, 350n7; arrival of, 1841; Cartesian coordinates and, 32; chain rule, 39, 267, 383n6, 389n22, 391n7; convergence and, 31, 15355, 161, 172; curl and, 152 (see also curl); derivatives and, 20, 30 (see also derivatives); differential, 1821 (see also differential calculus); divergence and, 298 (see also divergence); du Châtelet and, 3839, 41; Einstein and, 19, 3536; Euclid and, 27; Galileo and, 27; geometry and, 2122, 26, 3134, 3941, 350nn7–8, 352n15; grad and, 152 (see also grad); gravity and, 20, 27, 3440; Greeks and, 22, 29; Hamilton and, 1820, 41, 65; Harriot and, 27, 29, 3132, 350nn7–8; imaginary numbers and, 21; integral, 2027, 31, 35 (see also integrals); inverse square law and, 34, 36; laws of motion and, 3435; Leibniz and, 2730, 3541, 349n5, 350n6; Maxwell and, 18, 35, 349nn1–2; method of exhaustion and, 2226; motion and, 27, 30, 3436, 39, 41; Newton and, 1820, 2641, 323, 350nn6–7, 351nn11–13, 352n15; notation and, 29, 32, 3940; planetary motion and, 3436; Principia and, 3441, 350n7, 351nn12–13, 352n15; Pythagoras’s theorem and, 22, 23; slope, 40, 152, 383n6; symbolic, 39, 54, 83; tangent, 36, 222, 22526, 31516, 383n6; tensor, 17 (see also tensor calculus); velocity and, 32, 41, 352n15; Wallis and, 27, 3134, 350nn7–8; Zeno and, 2631, 40, 44

坎贝尔刘易斯:Maxwell和,119,150,153,157,166-67 Tait 119,150,167​

Campbell, Lewis: Maxwell and, 119, 150, 153, 157, 16667; Tait and, 119, 150, 167

坎农,安妮·贾普,361 n26

Cannon, Annie Jump, 361n26

电容器,12425,369 n20

capacitors, 12425, 369n20

Cardano , Girolamo:代数和,1116,347;算法,11 – 16,56,328,347 n16;大11,328,347 n16 背景,12 积分32,37几何和,1112,32,347 n16 塔尔塔利亚和,12,347 n16 时间轴328

Cardano, Girolamo: algebra and, 1116, 347; algorithm of, 1116, 56, 328, 347n16; Ars Magna, 11, 328, 347n16; background of, 12; calculus and, 32, 37; geometry and, 1112, 32, 347n16; Tartaglia and, 12, 347n16; timeline on, 328

埃利·嘉当, 31819 , 322 , 336 , 402 n23

Cartan, Élie, 31819, 322, 336, 402n23

笛卡尔坐标:微积分和,32;坐标变换,190另请参阅坐标变换);弯曲空间和,23033;爱因斯坦和,xxiv207400 n8;麦克斯韦和,128143;新维度和,xixxxvi;符号和,xxiixxiii;四元数和,150184186;空间和,7782;时空和,190;张量和,26227072284290295391 n7,392 n14 时间轴,328

Cartesian coordinates: calculus and, 32; coordinate transformations, 190 (see also coordinate transformations); curved space and, 23033; Einstein and, xxiv, 207, 400n8; Maxwell and, 128, 143; new dimensions and, xixxxvi; notation and, xxiixxiii; quaternions and, 150, 184, 186; space and, 77, 82; space-time and, 190; tensors and, 262, 27072, 284, 290, 295, 391n7, 392n14; timeline on, 328

天主教徒,48 , 50 , 116 , 240 , 243

Catholics, 48, 50, 116, 240, 243

柯西,奥古斯丁,131,213,215,230,239,332,350 n6

Cauchy, Augustin, 131, 213, 215, 230, 239, 332, 350n6

凯莱阿瑟:背景,8 ​​1 – 83,188 弯曲空间和,23031;曲面231,242;汉密尔顿8,8183,86,90,137,179,332不变性 191 94,255矩阵82 – 86,89 – 90,230,332 麦克斯韦,137,231 ;肖像 231 和,8,8183,86,90,179,187,194,206,332 33,359 n15 空间81 – 90 ;时空188 – 94,206,208 泰特 和,137,156,160,163,179,187,189,191,194,206,333,378 n3时间轴开启,33133

Cayley, Arthur: background of, 8183, 188; curved space and, 23031; curved surfaces and, 231, 242; Hamilton and, 8, 8183, 86, 90, 137, 179, 332; invariance and, 19194, 255; matrices and, 8286, 8990, 230, 332; Maxwell and, 137, 231; portrait of, 231; quaternions and, 8, 8183, 86, 90, 179, 187, 194, 206, 33233, 359n15; space and, 8190; space-time and, 18894, 206, 208; Tait and, 137, 156, 160, 163, 179, 187, 189, 191, 194, 206, 333, 378n3; timeline on, 33133

欧洲核子研究中心,320,402 n1

CERN, 320, 402n1

链式法则,39,267,383 n6,389 n22,391 n7​​

chain rule, 39, 267, 383n6, 389n22, 391n7

夏特莱,艾米丽·杜,3741 , 329

Châtelet, Émilie du, 3741, 329

ChatGPT, 249 , 387 nn11–12

ChatGPT, 249, 387nn11–12

中国数学xviixxvi9,82,85,323,327

Chinese mathematics, xvii, xxvi, 9, 82, 85, 323, 327

奇泽姆,格雷斯188 - 89,206,242,321,334

Chisholm, Grace, 18889, 206, 242, 321, 334

基督徒,47 , 72 , 133 , 137

Christians, 47, 72, 133, 137

Christoffel ,Elwin :弯曲空间和,23739 量和,257,27173,300,303,386 n15,397 n40,397 n42

Christoffel, Elwin: curved space and, 23739; tensors and, 257, 27173, 300, 303, 386n15, 397n40, 397n42

克莱鲁特,亚历克西斯-克劳德, 3839

Clairut, Alexis-Claude, 3839

Clarendon Press: Maxwell 和,157

Clarendon Press: Maxwell and, 157

Clebsch,Alfred222,256

Clebsch, Alfred, 222, 256

克利福德,露西163,167

Clifford, Lucy, 163, 167

克利福德,威廉·金登:弯曲空间和,230238;逝世,167动态几何原本161;乔治·艾略特和,163;格拉斯曼和,16162;赫斯特尼斯论,374 n23;讲座和论文166;麦克斯韦和,16367;符号和,16162关于几何基础的假设(黎曼译本),238;四元数和,16067172177179181;黎曼和,230238;时间轴论,333

Clifford, William Kingdon: curved space and, 230, 238; death of, 167; Elements of Dynamic, 161; George Eliot and, 163; Grassmann and, 16162; Hestenes on, 374n23; Lectures and Essays, 166; Maxwell and, 16367; notation and, 16162; On the Hypotheses Which Lie at the Bases of Geometry (translation of Riemann), 238; quaternions and, 16067, 172, 177, 179, 181; Riemann and, 230, 238; timeline on, 333

科恩·I·伯纳德,38岁

Cohen, I. Bernard, 38

柯勒律治,塞缪尔·泰勒,62 , 64

Coleridge, Samuel Taylor, 62, 64

碰撞:哈里奥特开创性矢量研究,5052,328,353 n8

collisions: Harriot’s pioneering vectorial study of, 5052, 328, 353n8

彩色摄影:麦克斯韦和,158,248 ;用于图像处理的张量和,248

colour photography: Maxwell and, 158, 248; tensors for image processing and, 248

度量衡委员会和公制105,329

Committee on Weights and Measures and the metric system, 105, 329

交换律:代数2,61,332,356 n2 ;密码,89,360 n23;爱因斯坦和,315广义相对论和,81,315格拉斯曼和,104,107,114 – 15 汉密尔顿2,61,76,79,90,99,104,107,114 16,163,178,332,345 n9,359 n13 矩阵 85 量子力学和,80 81 和,76,104,107,114-16,162-63,178,332,356n2;空间旋转 90 ; 245,268,286 ;向量,79

commutative law: algebra and, 2, 61, 332, 356n2; cryptography and, 89, 360n23; Einstein and, 315; general relativity and, 81, 315; Grassmann and, 104, 107, 11415; Hamilton and, 2, 61, 76, 79, 90, 99, 104, 107, 11416, 163, 178, 332, 345n9, 359n13; matrices and, 85; quantum mechanics and, 8081; quaternions and, 76, 104, 107, 11416, 16263, 178, 332, 356n2; rotations in space and, 90; tensors and, 245, 268, 286; vectors and, 79

完成平方,6,10-11,327,346 n9,347 n13,354 n11​​​

completing the square, 6, 1011, 327, 346n9, 347n13, 354n11

复数:代数16,348 n20 ;弯曲空间233欧拉和,5557汉密尔顿和,54,57,6064,67,69,7475,78,100,104,107,109,114,116,178,244,355 n20,356 n3,358 n11;向量概念和,53 63,66 – 67,355 n14,14 (参见八元和,99,363 n34 数和,104、107、109、114、116、178 量子比特和,251 52 平面上表示,58 60 ;旋转和,91 92,360 n24量和,244、251 – 52 时间轴328、330

complex numbers: algebra and, 16, 348n20; curved space and, 233; Euler and, 5557; Hamilton and, 54, 57, 6064, 67, 69, 7475, 78, 100, 104, 107, 109, 114, 116, 178, 244, 355n20, 356n3, 358n11; ideas for vectors and, 5363, 6667, 355n14; imaginary part of, 14 (see also imaginary numbers); octonions and, 99, 363n34; quaternions and, 104, 107, 109, 114, 116, 178; qubits and, 25152; representing on number plane, 5860; rotations and, 9192, 360n24; tensors and, 244, 25152; timeline on, 328, 330

复平面,5862 , 67 , 71 , 91 , 114 , 354 n11

complex plane, 5862, 67, 71, 91, 114, 354n11

守恒定律:Harriot 和,3534 n8;Leibniz 和,36;Newton 和,36;牛顿引力和,368 n15;Noether 和,16另见能量动量守恒定律)

conservation laws: Harriot and, 3534n8; Leibniz and, 36; Newton and, 36; Newtonian gravity and, 368n15; Noether and, 16 (see also conservation of energy-momentum)

能量动量守恒定律291,29930 0,30610,321,393 n21,400 n8 Bianchi恒等式31011;Noether和306 – 11,335,399 n1,399 – 40 0n5

conservation of energy-momentum, 291, 299300, 30610, 321, 393n21, 400n8; Bianchi identities and, 31011; Noether and, 30611, 335, 399n1, 399400n5

收敛:无穷级数的收敛,31 矢量场的收敛,15355、161、172、372 n7

convergence: of infinite series, 31; of vector fields, 15355, 161, 172, 372n7

坐标变换弯曲空间227,23437,383 n6 ;爱因斯坦307;框架19091;不变性和 190 – 94,2024,237,256,265,26869,305,307,389 n22,390 n23 时空19094,199 – 20 4,209​张量242,25556,25965,26869,272,283,287,291,295,389 n22,390 n23,392 n14,394 n30 时间轴开启,336​​​

coordinate transformations: curved space and, 227, 23437, 383n6; Einstein and, 307; frames and, 19091; invariance and, 19094, 2024, 237, 256, 265, 26869, 305, 307, 389n22, 390n23; space-time and, 19094, 199204, 209; tensors and, 242, 25556, 25965, 26869, 272, 283, 287, 291, 295, 389n22, 390n23, 392n14, 394n30; timeline on, 336

Corry ,Leo 297,395 n33

Corry, Leo, 297, 395n33

库仑,奥古斯丁:电与电,125130138329;定律,138154367 n11,370 n21;麦克斯韦关于库仑定律,138154367 n11,369 n19,370 n21 ;库仑定律矢量形式,154174371 n21

Coulomb, Augustin: electricity and, 125, 130, 138, 329; law of, 138, 154, 367n11, 370n21; Maxwell on Coulomb’s law, 138, 154, 367n11, 369n19, 370n21; vector form of Coulomb’s law, 154, 174, 371n21

方差(形式不变方程和相对论205,291,295,3045,392 n14,393 n21,394 n30,399 n2,400 n8

covariance (form-invariant equations and relativity), 205, 291, 295, 3045, 392n14, 393n21, 394n30, 399n2, 400n8

数,27172,300,30910,316,318,397 n40​

covariant derivatives, 27172, 300, 30910, 316, 318, 397n40

克伦威尔,奥利弗,27岁

Cromwell, Oliver, 27

交叉积:麦克斯韦和,144 四元数和,104、106、155、162、179 80 空间和,78 80 时空和,213 张量和,245

cross products: Maxwell and, 144; quaternions and, 104, 106, 155, 162, 17980; space and, 7880; space-time and, 213; tensors and, 245

克罗,迈克尔196,366 n19

Crowe, Michael, 196, 366n19

密码学,89,360 n23

cryptography, 89, 360n23

晶体,18;双轴,20;铁磁,363 n31;汉密尔顿和,20;雪,193;应力和,21324445;张量和,322;沃格特和,24445

crystals, 18; biaxial, 20; ferromagnetic, 363n31; Hamilton and, 20; snow, 193; stress forces and, 213, 24445; tensors and, 322; Voigt and, 24445

三次方程:代数和,1016,347 n15,348 n18 ;向量概念和,46,53,56时间轴328

cubic equations: algebra and, 1016, 347n15, 348n18; ideas for vectors and, 46, 53, 56; timeline on, 328

楔形文字,xiii - xiv10,265,327,343 n1,347 n13​​

cuneiform script, xiiixiv, 10, 265, 327, 343n1, 347n13

curl 麦克斯韦和,142,152,156 – 57,172,174,204,243,287,332,376 n10 ;四和,152,156 57,17275,370 n21,373 n11,376 n12,376 n14 ;时空和204,211,214;张和,243,271,287 时间轴开启,332

curl: Maxwell and, 142, 152, 15657, 172, 174, 204, 243, 287, 332, 376n10; quaternions and, 152, 15657, 17275, 370n21, 373n11, 376n12, 376n14; space-time and, 204, 211, 214; tensors and, 243, 271, 287; timeline on, 332

弯曲空间/时空:代数和,229,233 – 35,238阿基米德225;天文学和,22829;笛卡尔坐标和,23033 ;凯莱23031;克利福德和,230,238 复数和,233 坐标变换和,227,234 – 37,383 n6 导数和224,229,235 39 积分和,383 n5,385 n11 爱因斯坦和xii21719,226,228,230,232,23739,277,278,283,383 n5 电磁 237 ;欧几里得几何,220 – 22,22731,23435,239;欧拉和223,229 四维数学219,223,226;高斯21935,383 n4,383 n6,385 n8 ;​广义相对论和,2171922823827728283;测地线和,229;几何和,2202242622931238383 n6 ;格拉斯曼和,23033;引力和,219237385 n7 ;格罗斯曼和,2172023023839;汉密尔顿和,22930; Harriot 和, 221 , 22428 , 384 n6; 无穷小和, 22124 , 383 n6; 积分和, 225 , 383 n6; 不变性和, 22630 , 23538 ; 运动定律和, 230 ; Leibniz 和, 223 ; Levi-Civita 和, 21920 , 237 ; Lorentz 和, 227 ; 矩阵和, 230 , 384 n6; Maxwell 和, 219 , 231 , 23539 ;夫斯基和,217,219,223,227,234,238-39,383 n5,386 n16 运动217,219,230 牛顿,231 ;符号,​​​

curved space/space-time: algebra and, 229, 23335, 238; Archimedes and, 225; astronomy and, 22829; Cartesian coordinates and, 23033; Cayley and, 23031; Clifford and, 230, 238; complex numbers and, 233; coordinate transformations and, 227, 23437, 383n6; derivatives and, 224, 229, 23539; differential calculus and, 383n5, 385n11; Einstein and, xii, 21719, 226, 228, 230, 232, 23739, 277, 278, 283, 383n5; electromagnetism and, 237; Euclidean geometry and, 22022, 22731, 23435, 239; Euler and, 223, 229; four-dimensional mathematics and, 219, 223, 226; Gauss and, 21935, 383n4, 383n6, 385n8; general theory of relativity and, 21719, 228, 238, 27728, 283; geodesics and, 229; geometry and, 220, 22426, 22931, 238, 383n6; Grassmann and, 23033; gravity and, 219, 237, 385n7; Grossmann and, 21720, 230, 23839; Hamilton and, 22930; Harriot and, 221, 22428, 384n6; infinitesimals and, 22124, 383n6; integrals and, 225, 383n6; invariance and, 22630, 23538; laws of motion and, 230; Leibniz and, 223; Levi-Civita and, 21920, 237; Lorentz and, 227; matrices and, 230, 384n6; Maxwell and, 219, 231, 23539; Minkowski and, 217, 219, 223, 227, 234, 23839, 383n5, 386n16; motion and, 217, 219, 230; Newton and, 231; notation and,

弯曲空间/时空22223 , 23339 , 383 n6 ; 平行四边形规则和, 220 ; 庞加莱和, 218 ; 毕达哥拉斯定理和, 231 ; 二次方程和, 23338 ; 四元数和, 229 ; 里奇和, 21920 , 226 , 23739 ; 旋转和, 227 ; 标量和, 22526 , 234 , 236 , 383 n6, 386 n13 ; 索末菲和, 219 ; 狭义相对论和, 232 ; 对称性和, 236 ; 泰特和, 385 n12;张量微积分和,219;汤姆森和,236;三维空间和,219-23,229 矢量和,237;速度和,230

curved space/space-time 22223, 23339, 383n6; parallelogram rule and, 220; Poincaré and, 218; Pythagoras’s theorem and, 231; quadratic equations and, 23338; quaternions and, 229; Ricci and, 21920, 226, 23739; rotation and, 227; scalar numbers and, 22526, 234, 236, 383n6, 386n13; Sommerfeld and, 219; special theory of relativity and, 232; symmetry and, 236; Tait and, 385n12; tensor calculus and, 219; Thomson and, 236; three-dimensional space and, 21923, 229; vector field and, 237; velocity and, 230

曲面凯莱和,231,242 爱因斯坦和,314-15 高斯和,111,220-35,269,330,383 n6 曲面一般研究(高斯)221,383 n6 和,111,115 黎曼,219,230-36,244,269,383 n6,385 nn10-12,386 nn13-16包括曲率条件244,260,283,287时间轴 330,335

curved surfaces: Cayley and, 231, 242; Einstein and, 31415; Gauss and, 111, 22035, 269, 330, 383n6; General Investigations of Curved Surfaces (Gauss), 221, 383n6; quaternions and, 111, 115; Riemann and, 219, 23036, 244, 269, 383n6, 385nn10–12, 386nn13–16; tensors (incl. condition for curvature) and, 244, 260, 283, 287; timeline on, 330, 335

达朗贝尔,让·勒朗,57岁

d’Alembert, Jean Le Rond, 57

暗能量,322,397 n44

dark energy, 322, 397n44

暗物质,337,397 n44

dark matter, 337, 397n44

查尔斯·达尔文164

Darwin, Charles, 164

数据表示矩阵81,8689;美索不达米亚xiii xiv ,81,323,327量和 xi xiii19192,244 – 49,253 – 54,290,323;向量xi xiiixxv77,8689,19192

data representation: matrices and, 81, 8689; Mesopotamian, xiii–xiv, 81, 323, 327; tensors and, xixiii, 19192, 24449, 25354, 290, 323; vectors and, xixiii, xxv, 77, 8689, 19192

DeepMind 191-92

DeepMind, 19192

德·摩根,奥古斯都:代数结构和,99 – 100,163,183背景72复数 63 死亡,151;家庭生活,73;汉密尔顿和,104,106,112 – 14 向量思想和,6163,68 ;洛夫莱斯和,73,358 n8 宗派观点72 – 72,85,160

De Morgan, Augustus: algebraic structure and, 99100, 163, 183; background of, 72; complex numbers and, 63; death of, 151; family life of, 73; Hamilton and, 104, 106, 11214; ideas for vectors and, 6163, 68; Lovelace and, 73, 358n8; nonsectarian views and, 7272, 85, 160

德·摩根,索菲亚·弗伦德,7374

De Morgan, Sophia Frend, 7374

导数:协变,2717230030910316318397 n40;弯曲空间和,22422923539;爱因斯坦和,30910316318;几何和,3940;汉密尔顿和,204163152298;向量的思想和,63;莱布尼茨和,39368 n12;麦克斯韦和,1262713738141152172264271300369 n21,376 n14 ;牛顿和,63 137172182202271295301352 n15,368 n12,373 n11;部分,12627 2352642717228889298300,368 n13,389 n22,397 n40 和,152,17275,182;时空 202,345 n8 264,27172,283,286,28889,295,298 – 30 1,30912,386 n9,389 n22,390 n23,397 n40;时间轴335 芝诺和,30​​​

derivatives: covariant, 27172, 300, 30910, 316, 318, 397n40; curved space and, 224, 229, 23539; Einstein and, 30910, 316, 318; geometry and, 39, 40; Hamilton and, 20, 41, 63, 152, 298; ideas for vectors and, 63; Leibniz and, 39, 368n12; Maxwell and, 12627, 13738, 141, 152, 172, 264, 271, 300, 369n21, 376n14; Newton and, 63, 137, 172, 182, 202, 271, 295, 301, 352n15, 368n12, 373n11; partial, 12627, 235, 264, 27172, 28889, 298, 300, 368n13, 389n22, 397n40; quaternions and, 152, 17275, 182; space-time and, 202, 345n8; tensors and, 264, 27172, 283, 286, 28889, 295, 298301, 30912, 386n9, 389n22, 390n23, 397n40; timeline for, 335; Zeno and, 30

笛卡尔,勒内,xix;代数和,6915350 nn7–8;解析几何和,32;笛卡尔坐标和,32112另请参阅笛卡尔坐标);方法论9 , 328;向量的思想和,xix5052;虚数和,61552346 n8;麦克斯韦,151;时间轴,328

Descartes, René, xix; algebra and, 6, 9, 15, 350nn7–8; analytic geometry and, 32; Cartesian coordinates and, 32, 112 (see also Cartesian coordinates); Discourse on Method, 9 328; ideas for vectors and, xix, 50, 52; imaginary numbers and, 6, 15, 52, 346n8; Maxwell on, 151; timeline on, 328

狄更斯,查尔斯:泰特和,149

Dickens, Charles: Tait and, 149

微分学:弯曲空间和,219,22223,226,23138,383 n5,385 n11 发展,18 21,26 – 29,35,40,328,333 – 34,336,350 n6,352 n15 爱因斯坦 318 ;向量思想,62 麦克斯韦127,13741,369 n19,369 n21 和,105,109,111,371 n23 变化率和,2021​张量和,24244,256 – 58,268 – 74,284,289,389 n22,393 n19 时间轴开启328,333 – 34,336

differential calculus: curved space and, 219, 22223, 226, 23138, 383n5, 385n11; development of, 1821, 2629, 35, 40, 328, 33334, 336, 350n6, 352n15; Einstein and, 318; ideas for vectors and, 62; Maxwell and, 127, 13741, 369n19, 369n21; quaternions and, 105, 109, 111, 371n23; rates of change and, 2021; tensors and, 24244, 25658, 26874, 284, 289, 389n22, 393n19; timeline on, 328, 33334, 336

微分几何:Cartan 和,322336;Gauss 和,10922323132243269272369 n21;Grassmann 和,109;向量的思想和,352 n15;张量和,272289393 n19

differential geometry: Cartan and, 322, 336; Gauss and, 109, 223, 23132, 243, 269, 272, 369n21; Grassmann and, 109; ideas for vectors and, 352n15; tensors and, 272, 289, 393n19

衍射:干涉图案19,97,199,363 n31;光和,19,27778,293,302 – 3,317,335,337,397 n44

diffraction: interference patterns and, 19, 97, 199, 363n31; light and, 19, 27778, 293, 3023, 317, 335, 337, 397n44

丢番图,7

Diophantus, 7

狄拉克,保罗爱因斯坦和,402 n2;电子和,96,155,32021,336,362 n30 ;单极子和,155,376 n13;矩阵和,96,320,323,362 n29 诺贝尔奖,402 n2 符号和,251 泡利自旋矩阵和,336;的QED方程320量子理论和,96,155,251,32021,336,362 n29,376 n13,402 n2 空间 96,362 n29 狭义相对论和,320;张量和,25153,323;矢量和251

Dirac, Paul: Einstein and, 402n2; electrons and, 96, 155, 32021 336, 362n30; magnetic monopoles and, 155, 376n13; matrices and, 96, 320, 323, 362n29; Nobel Prize of, 402n2; notation and, 251; Pauli spin matrices and, 336; QED equations of, 320; quantum theory and, 96, 155, 251, 32021, 336, 362n29, 376n13, 402n2; space and, 96, 362n29; special theory of relativity and, 320; tensors and, 25153, 323; vectors and, 251

方法论》(笛卡尔),9,328

Discourse on Method (Descartes), 9, 328

迪士尼,凯瑟琳(汉密尔顿和)73,145

Disney, Catherine (Hamilton and), 73, 145

分配律255,356 n2​

distributive law, 255, 356n2

发散(矢量和张场)爱因斯坦308,400 n6 麦克斯韦和,144,149,152 – 55,161,172,243,264,332,369 n21,372 n7,375 n10 15257,161,17274 时空 和 ,211,397 n39 ;243,264,29830 1 定理139,144,152,369 n21 时间轴,332

divergence (of vector and tensor fields): Einstein and, 308, 400n6; Maxwell and, 144, 149, 15255, 161, 172, 243, 264, 332, 369n21, 372n7, 375n10; quaternions and, 15257, 161, 17274; space-time and, 211, 397n39; tensors and, 243, 264, 298301; theorem of, 139, 144, 152, 369n21; timeline on, 332

查尔斯·道奇森( Lewis Carroll ),xxvi2,345 n9,346 n2

Dodgson, Charles (Lewis Carroll), xxvi, 2, 345n9, 346n2

多普勒效应,278

Doppler effect, 278

点积78,80,211

dot products, 78, 80, 211

都柏林大学(汉密尔顿关于四元数的讲座),82

Dublin University (Hamilton’s lectures on quaternions), 82

邓辛克天文台(汉密尔顿的家)1,73

Dunsink Observatory (Hamilton’s home), 1, 73

E = mc 2方程,920207287347 n12,402 n2

E = mc2 equation, 9, 20, 207, 287, 347n12, 402n2

诺森伯兰伯爵(哈里奥特的赞助人),48 , 50

Earl of Northumberland (Harriot’s patron), 48, 50

日食天文学和,98,199,294,302-3,317,320,335,397 n44;广义相对论和,294,302-3,317,320,335

eclipses: astronomy and, 98, 199, 294, 3023, 317, 320, 335, 397n44; general theory of relativity and, 294, 3023, 317, 320, 335

爱丁顿,阿瑟,317 , 335

Eddington, Arthur, 317, 335

埃奇沃思,弗朗西斯·博福特,62 , 64

Edgeworth, Francis Beaufort, 62, 64

埃奇沃思,玛丽亚,62 , 6465

Edgeworth, Maria, 62, 6465

埃及数学,xvixxvi阿美斯,2223、122、327、349 n3 代数和,9;生死循环,32;几何和,2223;希帕提娅,xvii7328;土地地块和,186

Egyptian mathematics, xvi, xxvi; Ahmes, 2223, 122, 327, 349n3; algebra and, 9; cycle of birth and death, 32; geometry and, 22, 23; Hypatia, xvii, 7, 328; land parcels and, 186

保罗·埃伦菲斯特,96、31112、362 n28、393 n23、395 n32​​

Ehrenfest, Paul, 96, 31112, 362n28, 393n23, 395n32

爱因斯坦,阿尔伯特:亚伯拉罕和,280 81,292,294,312,391 n10代数和,7,9,17,313,316,347 n12 算法和,313 – 14;反犹太主义 196;天文学和 200,293,322,336;伯尔尼大学232;“最大的错误” ,322 布朗运动和,347 n12 积分和,19,35 – 36 ;笛卡尔坐标xxiv207,400 n8克利福德相比,167 交换律和,315;能量守恒定律和,3069335393 n21,400 n8;动量守恒定律和,3059321;坐标变换和,283291295300305307 方差和,2052912953045392 n14,393 n21,394 n30,399 n2,400 n8 ;弯曲空间230 弯曲时空和xxii116,23839,277,278,28385;曲面和,31415;“文化世界宣言”和,293;导数和,30910316318;微积分和,318;狄拉克和,402 n2 ;散度和,308400 n6 ;博士学位和,21718;E = mc 2和,920207287347 n12 ,402 n2 ;电磁学与,308318;电子与,9200280380 n12 ;优雅方程,325设计理论与,29095393 n21 ;以太与,19899;欧几里得与,308;欧拉与,305;通量与,309广义相对论的基础305392 n12 ;参考系和,27127628384;广义相对论与,927430 3 (另请参阅广义相对论); Grassmann 和, 318 ;重力和, 3059 , 313 , 315 , 318 , 32122 , 399 nn4–5;Grossmann 和, 21719 , 23839 , 244 , 253 , 258 , 275 , 28395 , 29899 , 303 , 319 , 33334 ;Hamilton 和, 3067 , 310 , 315 ;希尔伯特和,293 98 301 304 6 310 313 317 18 335 395 nn33–35,396 nn36–37,397 n43;向量的思想和,6263;积分和,3056;不变性和,2262833058323399 n5;克莱因和304631013;拉格朗日和,3057;运动定律和,306;列维-奇维塔和,21931031519;光和,193514120020 1,2052122778529394303317335337347 n12,379 n10,397 n44;洛伦兹和,200;洛伦兹变换19921 2,242,280,283 – 84,334,380 n12,381 n14,392 n12,394 n30,399 n5;Marić和,19720 0,293,333,335,379 nn9–10,394 n31 ​​​​ Maxwell和xii7、35、129、141、160、183、196、200 – 20 1、205、207、212、238、241 – 42、280、287、289、325、334、366 n1、380 n12Minkowski196、207 – 12、217、219、238、242、280 – 87、289、293、303、334、340、383 n5 动量和,30511,321运动和,35,20020 1217219275295963068347 n12 ,379 n10 ,392 n12 ,395 n32 ,397 n44 ;牛顿万有引力定律和,2757627928328586;诺特和,30413317399 n3 ,399 n5 ;诺顿论,399 n2 ;符号和,30030 1 ;平行四边形规则和,314庞加莱和,2001,205 9,212,218,242,280 – 81,334 等效原理276 – 78,282 – 85量子力学311,320 – 21,323 量子理论19,292,302,311,347 n12180,207里奇219,305,310,314,316 – 19 黎曼和,31011;标和,401 n12 ;​​索末菲和,317;空间和,95;时空和,xxiv19621 2 ,21632122379 n10 ,381 n14 ,383 n5 ;狭义相对论和,160另见狭义相对论);对称性和,30614322399 n5 ;张量微积分和,28590305316318 ;和,xii160183241442532582612667127530 3,321390 n2,391 nn6–10,392 nn11–14,393 n21,394 nn30–31,395 nn32–35,396 n37,397 n38,397 nn41–44,399 n46 ;时间轴开启,329 37 速度,305 6

Einstein, Albert: Abraham and, 28081, 292, 294, 312, 391n10; algebra and, 7, 9, 17, 313, 316, 347n12; algorithms and, 31314; anti-Semitism and, 196; astronomy and, 200, 293, 322, 336; Bern University and, 232; “biggest blunder” of, 322; Brownian motion and, 347n12; calculus and, 19, 3536; Cartesian coordinates and, xxiv, 207, 400n8; Clifford compared with, 167; commutative law and, 315; conservation of energy and, 3069, 335, 393n21, 400n8; conservation of momentum and, 3059, 321; coordinate transformations and, 283, 291, 295, 300, 305, 307; covariance and, 205, 291, 295, 3045, 392n14, 393n21, 394n30, 399n2, 400n8; curved space and, 230; curved space-time and, xxii, 116, 23839, 277, 278, 28385; curved surfaces and, 31415; “Declaration to the Cultural World” and, 293; derivatives and, 30910, 316, 318; differential calculus and, 318; Dirac and, 402n2; divergence and, 308, 400n6; doctorate and, 21718; E = mc2 and, 9, 20, 207, 287, 347n12, 402n2; electromagnetism and, 308, 318; electrons and, 9, 200, 280, 380n12; elegant equation of, 325; Entwurf theory and, 29095, 393n21; ether and, 19899; Euclid and, 308; Euler and, 305; flux and, 309; The Foundation of the General Theory of Relativity, 305, 392n12; frames of reference and, 271, 276, 28384; general theory of relativity and, 9, 274303 (see also general theory of relativity); Grassmann and, 318; gravity and, 3059, 313, 315, 318, 32122, 399nn4–5; Grossmann and, 21719, 23839, 244, 253, 258, 275, 28395, 29899, 303, 319, 33334; Hamilton and, 3067, 310, 315; Hilbert and, 29398, 301, 3046, 310, 313, 31718, 335, 395nn33–35, 396nn36–37, 397n43; ideas for vectors and, 6263; integrals and, 3056; invariance and, 226, 283, 3058, 323, 399n5; Klein and, 3046, 31013; Lagrange and, 3057; laws of motion and, 306; Levi-Civita and, 219, 310, 31519; light and, 19, 35, 141, 200201, 205, 212, 27785, 29394, 303, 317, 335, 337, 347n12, 379n10, 397n44; Lorentz and, 200; Lorentz transformations and, 199212, 242, 280, 28384, 334, 380n12, 381n14, 392n12, 394n30, 399n5; Marić and, 197200, 293, 333, 335, 379nn9–10, 394n31; Maxwell and, xii, 7, 35, 129, 141, 160, 183, 196, 200201, 205, 207, 212, 238, 24142, 280, 287, 289, 325, 334, 366n1, 380n12; Minkowski and, 196, 20712, 217, 219, 238, 242, 28087, 289, 293, 303, 334, 340, 383n5; momentum and, 30511, 321; motion and, 35, 20020 1, 217, 219, 275, 29596, 3068, 347n12, 379n10, 392n12, 395n32, 397n44; Newton’s law of gravity and, 27576, 279, 283, 28586; Noether and, 30413, 317, 399n3, 399n5; Norton on, 399n2; notation and, 300301; parallelogram rule and, 314; Poincaré and, 2001, 2059, 212, 218, 242, 28081, 334; principle of equivalence and, 27678, 28285; quantum mechanics and, 311, 32021, 323; quantum theory and, 19, 292, 302, 311, 347n12; quaternions and, 180, 207; Ricci and, 219, 305, 310, 314, 31619; Riemann and, 31011; scalar numbers and, 401n12; Sommerfeld and, 317; space and, 95; space-time and, xxiv, 196212, 216, 32122, 379n10, 381n14, 383n5; special theory of relativity and, 160 (see also special theory of relativity); symmetry and, 30614, 322, 399n5; tensor calculus and, 28590, 305, 316, 318; tensors and, xii, 160, 183, 24144, 253, 258, 261, 26671, 275303, 321, 390n2, 391nn6–10, 392nn11–14, 393n21, 394nn30–31, 395nn32–35, 396n37, 397n38, 397nn41–44, 399n46; timeline on, 32937; velocity and, 3056

爱因斯坦,汉斯·阿尔伯特,296,403 n5

Einstein, Hans Albert, 296, 403n5

爱因斯坦-理论318,318-19

Einstein-Cartan theory, 318, 31819

爱因斯坦求和约定,261

Einstein summation convention, 261

费迪南·爱森斯坦,82 岁

Eisenstein, Ferdinand, 82

《电工》(杂志),170

Electrician, The (magazine), 170

电:安培和,11011128131138174330365 n14,369 n21,371 n23 电容,12425,369 n20;库仑和,125130138329;法拉第和,1282913342369 n21;作为流体,13940;通量和,1303313738369 nn20–21 ;产生,12931132155330 感应和,133,138,151,369 n21,375 n10 ;拉格朗日,125 – 26拉普拉斯和,127,130麦克斯韦 123 45,150,154,157,168,332,343 n2,349 n2,365 n14,367 n11,368 n14,369 n19,369 n21,372 n7,376 n14 自然效应 325 ;泊松127,130 ​​​/向量和,150,154,157,168-69,185,195 静态124,128-30,136,138,290,329,369n21;Tait和 125,127,143,145,150,185,195 ;时间轴开启,332 电压,125,152 157 , 172

electricity: Ampère and, 11011, 128, 131, 138, 174, 330, 365n14, 369n21, 371n23; capacitance, 12425, 369n20; Coulomb and, 125, 130, 138, 329; Faraday and, 12829, 13342, 369n21; as fluid, 13940; flux and, 13033, 13738, 369nn20–21; generation of, 12931, 132, 155, 330; induction and, 133, 138, 151, 369n21, 375n10; Lagrange and, 12526; Laplace and, 127, 130; Maxwell and, 12345, 150, 154, 157, 168, 332, 343n2, 349n2, 365n14, 367n11, 368n14, 369n19, 369n21, 372n7, 376n14; natural effects of, 325; Poisson and, 127, 130; quaternions/vectors and, 150, 154, 157, 16869, 185, 195; static, 124, 12830, 136, 138, 290, 329, 369n21; Tait and, 125, 127, 143, 145, 150, 185, 195; timeline on, 332; voltage and, 125, 152, 157, 172

脑电图(EEG),247

electroencephalograms (EEGs), 247

电磁理论Heaviside), 170,176-77

Electromagnetic Theory (Heaviside), 170, 17677

电磁波:代数和,16,348 n17;麦克斯韦和,141 元数和,151,159,168,17172,371 n23 量和,289 – 90 ;时间轴332

electromagnetic waves: algebra and, 16, 348n17; Maxwell and, 141; quaternions and, 151, 159, 168, 17172, 371n23; tensors and, 28990; timeline for, 332

电磁学:超距作用13233,135,138,150,203,365 n14,369 n19,376 n11代数3,16,348 n17 弯曲空间和,237 爱因斯坦308,318以太和,280 ;亥赛和,17276;感应133,138,151,369 n21,375 n10 铁屑和95,13440,174,183;光和 141 参见麦克斯韦3,16,110,11846,150 – 51,156 – 60,17073,177,183,19820 4,213,241,271,280,288 – 90,320,325,332 33,371 n23,376 n11;单极子和155,372 n9,376 n13,385 n7 ​​​ ​Øersted 和, 94 , 110 , 124 , 12829 , 132 , 138 , 173 , 330 ; 四元数/向量和, 110 , 146 , 15051 , 15660 , 168 , 17073 , 17677 , 183 , 185 , 371 n23, 372 n9, 373 n11, 376 n11, 376 n13; 空间和, 8384 , 9899 ; 时空和, 19820 4, 213 ; Tait 和,118 24、128 29、137、139、142 45 ;电报95,2412442712752802889029699391 n8、397 n44 时间轴、330 332 33 电磁麦克斯韦)、145150154157168332343 n2、349 n2、365 n14、367 n7、367 n11、369 n19、369 n21、372 n7、373 n12;矢量3、11845

electromagnetism: action-at-a-distance and, 13233, 135, 138, 150, 203, 365n14, 369n19, 376n11; algebra and, 3, 16, 348n17; curved space and, 237; Einstein and, 308, 318; ether and, 280; Heaviside and, 17276; induction and, 133, 138, 151, 369n21, 375n10; iron filings and, 95, 13440, 174, 183; light and, 141 (see also light); Maxwell and, 3, 16, 110, 11846, 15051, 15660, 17073, 177, 183, 198204, 213, 241, 271, 280, 28890, 320, 325, 33233, 371n23, 376n11; monopoles and, 155, 372n9, 376n13, 385n7; Øersted and, 94, 110, 124, 12829, 132, 138, 173, 330; quaternions/vectors and, 110, 146, 15051, 15660, 168, 17073, 17677, 183, 185, 371n23, 372n9, 373n11, 376n11, 376n13; space and, 8384, 9899; space-time and, 198204, 213; Tait and, 11824, 12829, 137, 139, 14245; telegraphy and, 95; tensors and, 241, 244, 271, 275, 280, 28890, 29699, 391n8, 397n44; timeline for, 330, 33233; Treatise on Electricity and Magnetism (Maxwell), 145, 150, 154, 157, 168, 332, 343n2, 349n2, 365n14, 367n7, 367n11, 369n19, 369n21, 372n7, 373n12; vector field and, 3, 11845

电动势强度,153

electromotive intensity, 153

电子动量,94,96,335-36 库仑定律,154,174,371 n21;狄拉克和,96,155,320-21,336,362 n30 发现165爱因斯坦9,200,280,380 n12 ;矢量的想法和,55,66干涉图案和,19,55,363 n31;洛伦兹和 96,175,200,280,362 n28,380 n12 麦克斯韦和,124-25,129​单极子和,155;Opat 和,336363 n31;四元数/矢量和,175;薛定谔方程和,348 n17;空间和,93100;时空和,200;谱线和,361 n26;自旋,949817525129231832133536362 nn27–30,363 n31;量和,251280

electrons: angular momentum of, 94, 96, 33536; Coulomb’s law for, 154, 174, 371n21; Dirac and, 96, 155, 32021, 336, 362n30; discovery of, 165, Einstein and, 9, 200, 280, 380n12; ideas for vectors and, 55, 66; interference patterns and, 19, 55, 363n31; Lorentz and, 96, 175, 200, 280, 362n28, 380n12; Maxwell and, 12425, 129; monopoles and, 155; Opat and, 336, 363n31; quaternions/vectors and, 175; Schrödinger’s equation and, 348n17; space and, 93100; space-time and, 200; spectral lines and, 361n26; spin of, 9498, 175, 251, 292, 318, 321, 33536, 362nn27–30, 363n31; tensors and, 251, 280

四元数初等论述 Tait),14750,332

Elementary Treatise on Quaternions (Tait), 14750, 332

几何原本》(欧几里得)xviii4-5,327

Elements (Euclid), xviii, 45, 327

动态元素(Clifford),161

Elements of Dynamic (Clifford), 161

向量分析的元素(吉布斯),17880

Elements of Vector Analysis (Gibbs), 17880

乔治·艾略特,16366,230

Eliot, George, 16366, 230

拉尔夫·沃尔多·爱默生84岁

Emerson, Ralph Waldo, 84

大英百科全书133,380 n11​

Encyclopedia Brittanica, 133, 380n11

Entwurf理论,29095,393 n21

Entwurf theory, 29095, 393n21

埃拉托色尼,327

Eratosthenes, 327

动力学论文(莱布尼茨),4445

Essay on Dynamics (Leibniz), 4445

以太:爱因斯坦和,198 20 0,205,280;电磁学和,198,201 洛伦兹和,205,292;麦克斯韦和,198,380 n11;迈克尔逊-莫雷实验和,19899 行星运动假说和,3435 庞加莱和,20020 1,205,280,334 ;狭义相对论19820 1,205

ether: Einstein and, 198200, 205, 280; electromagnetism and, 198, 201; Lorentz and, 205, 292; Maxwell and, 198, 380n11; Michelson-Morley experiment and, 19899; planetary motion hypothesis and, 3435; Poincaré and, 200201, 205, 280, 334; special theory of relativity and, 198201, 205

欧几里得:代数和,45;阿波罗尼乌斯和,xix;微积分和,27;弯曲空间和,220222273123435239;爱因斯坦与非欧几里得几何,308元素xviii45327;平面空间与,115 17几何与,4,74,11517,163,210,216,220,22931,303,308,327,330空间与,74 ;时空20910,216欧几里得空间268 – 72,299 张量欧几里得空间284见广义相对论);时间轴327,330 欧几里得空间中向量,253,264 – 65

Euclid: algebra and, 45; Apollonius and, xix; calculus and, 27; curved space and, 22022, 22731, 23435, 239; Einstein and non-Euclidean geometry, 308; Elements, xviii, 45, 327; flat space and, 11517, geometry and, 4, 74, 11517, 163, 210, 216, 220, 22931, 303, 308, 327, 330; space and, 74; space-time and, 20910, 216; tensors and Euclidean spaces, 26872, 299; tensors and non-Euclidean spaces, 284 (see also general theory of relativity); timeline on, 327, 330; vectors in Euclidean space, 253, 26465

克尼多斯的欧多克索斯,344 n6

Eudoxus of Cnidus, 344n6

欧拉,莱昂哈德,xiv;代数和,348 n20;背景,54;美丽的公式,5556325;复数和,5557;弯曲空间和,223229;爱因斯坦和,305;流体流动和,136;向量的思想和,52576067354 nn12–13,355 nn14–15;虚数和,5257;记忆/失明,54;现代化牛顿和,105 空间和,728190356 n3,360 n24

Euler, Leonhard, xiv; algebra and, 348n20; background of, 54; beautiful formula of, 5556, 325; complex numbers and, 5557; curved space and, 223, 229; Einstein and, 305; fluid flow and, 136; ideas for vectors and, 5257, 60, 67, 354nn12–13, 355nn14–15; imaginary numbers and, 5257; memory/blindness of, 54; modernising Newton and, 105; space and, 72, 81, 90, 356n3, 360n24

幼发拉底河,xiv

Euphrates River, xiv

事件视界望远镜,337

Event Horizon Telescope, 337

因式分解:复数和,54;矩阵(图像压缩)和,86;多项式(Harriot)和15,348 n19

factorisation: complex numbers and, 54; matrices (image compression) and, 86; polynomials (Harriot) and, 15, 348n19

落下运动,112,125,275-76,308,353n6

falling motion, 112, 125, 27576, 308, 353n6

法拉第,迈克尔:电与,1282913342369 n21;法拉第定律,174;法拉第张量,289;场与,13436135279;麦克斯韦与,1282913342149151167174237279289331369 n21;Øersted 与,12829173330;“论法拉第力线”(麦克斯韦),136;Riebau 与,13334;Tait 与,149; Thomson 和,147、151 时间轴,330 31 矢量场和,13640

Faraday, Michael: electricity and, 12829, 13342, 369n21; Faraday’s law, 174; Faraday tensor, 289; fields and, 13436, 135, 279; Maxwell and, 12829, 13342, 149, 151, 167, 174, 237, 279, 289, 331, 369n21; Øersted and, 12829, 173, 330; “On Faraday’s Lines of Force” (Maxwell), 136; Riebau and, 13334; Tait and, 149; Thomson and, 147, 151; timeline on, 33031; vector field and, 13640

法西斯分子,317

Fascists, 317

联邦理工学院(或“Poly”,现为 ETH):Besso 和,281;Christoffel 和,238;Einstein 和,196218275281;Grossmann 和,21718;Marić 和,197;Minkowski 和,196

Federal Polytechnic School (or “Poly,” now ETH): Besso and, 281; Christoffel and, 238; Einstein and, 196, 218, 275, 281; Grossmann and, 21718; Marić and, 197; Minkowski and, 196

弗格森,邓肯(格伦莱尔123,367 n9

Ferguson, Duncan (Glenlair), 123, 367n9

费马,皮埃尔·德:定理,27,53,56-57,355 nn14-15;勒布朗,53,220 ;法律背景,52 Wallis 和 350 n8

Fermat, Pierre de: last theorem of, 27, 53, 5657, 355nn14–15; as Le Blanc, 53, 220; legal background of, 52; Wallis and, 350n8

Ferro,Scipione del,12

Ferro, Scipione del, 12

菲奥尔,安东尼奥,12岁

Fior, Antonio, 12

第一民族,xv

First Nations, xv

菲茨杰拉德,乔治,168 , 171 , 173 , 199

FitzGerald, George, 168, 171, 173, 199

《平面国:多维度的浪漫史》(Abbott),206

Flatland: A Romance of Many Dimensions (Abbott), 206

弗莱明,安布罗斯,168

Fleming, Ambrose, 168

通量:爱因斯坦和,309;电和,130 33,137 38,369 nn20–21; 向量的思想和,63 麦克斯韦和,13033,13738,369 nn20–21 数 /向量,151,154,155,174,279,375 n10 289 – 90

flux: Einstein and, 309; electricity and, 13033, 13738, 369nn20–21; ideas for vectors and, 63; Maxwell and, 13033, 13738, 369nn20–21; quaternions/vectors and, 151, 154, 155, 174, 279, 375n10; tensors and, 28990

Foote,Eunice Newton84,331,359 n19

Foote, Eunice Newton, 84, 331, 359n19

广义相对论的基础爱因斯坦305,392 n12

Foundation of the General Theory of Relativity, The (Einstein), 305, 392n12

“物理学基础”(希尔伯特),297

“Foundations of Physics” (Hilbert), 297

Fourier , Joseph :热流136,142,151,233,330,359 n19​

Fourier, Joseph: heat flow and, 136, 142, 151, 233, 330, 359n19

第四代数和,17 弯曲空间和,219,223,226 引力,381 n18 汉密尔顿和,2,4,17,7578,91,9899,206 立方体和,205 11 闵可夫斯基和,2078 四元和,187(参见);空间75 78,91,98 99,381 n16 时空17,75,187,20512,248,289,298,381 n16,381 n18 张量24748,28689,298;矢量分析和205 – 12

fourth dimension: algebra and, 17; curved space and, 219, 223, 226; gravity and, 381n18; Hamilton and, 2, 4, 17, 7578, 91, 9899, 206; hypercubes and, 20511; Minkowski and, 2078; quaternions and, 187 (see also quaternions); space and, 7578, 91, 9899, 381n16; space-time and, 17, 75, 187, 20512, 248, 289, 298, 381n16, 381n18; tensors and, 24748, 28689, 298; vector analysis and, 20512

第四名牧马人(乔治·格林),126

Fourth Wrangler (George Green), 126

参考系:守恒定律和,3068;坐标变换和,85194199201400 n10;定义,19091;惯性,28384;不变性和,18919 3 ,2024227255;相对论和,201205276, 277 , 278284291302305,336 张量和,259 60262271288290 295

frames of reference: conservation laws and, 3068; coordinate transformations and, 85, 194, 199, 201, 400n10; definition of, 19091; inertial, 28384; invariance and, 189193, 2024, 227, 255; relativity and, 201, 205, 276, 277, 278, 284, 291, 302, 305, 336; tensors and, 25960, 262, 271, 288, 290, 295

法国科学院105,126

French Academy of Sciences, 105, 126

法国大革命,105

French Revolution, 105

弗里德曼,亚历山大,322

Friedmann, Alexander, 322

模糊逻辑, 87

fuzzy logic, 87

伽利略:弹道学(抛物线轨迹)和,49,277,353 n6 积分和,27 ;运动和,49,112,275,390 n2,393 n21 矢量思想和4649,353 n6 相对论201,275

Galileo: ballistics (parabolic trajectory) and, 49, 277, 353n6; calculus and, 27; falling motion and, 49, 112, 275, 390n2, 393n21; ideas for vectors and, 4649, 353n6; relativity and, 201, 275

伽罗瓦,埃瓦里斯特,208

Galois, Évariste, 208

高斯,卡尔·弗里德里希:创造力,11516 ; 复数和,1089 ; 库仑定律/通量和,154174279 ; 弯曲空间和,21935383 n4,383 n6,385 n8;曲面和,111 ,22035 269330383 n6 ;微分几何和,109 22323132243269272,369 n21;曲面的一般研究 221 383 n6;格拉斯曼和,1089向量思想53,58,60-61,67不变性,226-30 ;麦克斯韦126,130,133,138-39,154,279,369 n21 莫比乌斯和,109 ;非欧几里得几何,115 符号和,383 n6 著定理,225 黎曼和,219,230-33,235,243,269,303,332,383 n6空间和,82-83,359 n16 量和,243,269,272,284,303 时间轴 327,330,332​

Gauss, Carl Friedrich: creativity of, 11516; complex numbers and, 1089; Coulomb’s law/flux and, 154, 174, 279; curved space and, 21935, 383n4, 383n6, 385n8; curved surfaces and, 111, 22035, 269, 330, 383n6; differential geometry and, 109, 223, 23132, 243, 269, 272, 369n21; General Investigations of Curved Surfaces, 221, 383n6; Grassmann and, 1089; ideas for vectors and, 53, 58, 6061, 67; invariance and, 22630; Maxwell and, 126, 130, 133, 13839, 154, 279, 369n21; Möbius and, 109; non-Euclidean geometry and, 115; notation and, 383n6; remarkable theorem of, 225; Riemann and, 219, 23033, 235, 243, 269, 303, 332, 383n6; space and, 8283, 359n16; tensors and, 243, 269, 272, 284, 303; timeline on, 327, 330, 332

高斯-博内定理,228

Gauss-Bonnet theorem, 228

高斯-库仑定律,154,174,279

Gauss-Coulomb laws, 154, 174, 279

高斯元法,8283,327,359 n16

Gaussian elimination, 8283, 327, 359n16

曲面的一般研究高斯),221,383 n6

General Investigations of Curved Surfaces (Gauss), 221, 383n6

广义相对论代数9 确认,33637 弯曲空间和,21719,228,238 发展,308,313,318 – 19,399 n2,400 n8,401 n11 ;E = mc 2,9,20,207,287,347 n12,402 n2;日食证明294,3023,317,320,335Entwurf理论290 – 95,393 n21;​广义相对论的基础(爱因斯坦) ,305,392 n12 ;参考,271,276,283-84 Friedmann和322;引力和,219,275 – 76,280 – 81,291,302,308,313,318,329,335,372,395 n33,395 n35,397 n44,399 nn4–5 Grossmann210,217,258,275,291,298,303,319,33435 冲击320 22 Noether和,304 – 5 ;平行四边形规则314 19空间80 – 83,96 时空和19697,206,21012 和,160,258,267,274 – 30 3,323,372 n9,389 n20,391 n9,392 n12,392 n14,394 n39,395 nn32–35,397 n44 时间轴,329,33437 矢量160​​​​​

general theory of relativity: algebra and, 9; confirmation of, 33637; curved space and, 21719, 228, 238; development of, 308, 313, 31819, 399n2, 400n8, 401n11; E = mc2 and, 9, 20, 207, 287, 347n12, 402n2; eclipse proof of, 294, 3023, 317, 320, 335; Entwurf theory and, 29095, 393n21; The Foundation of the General Theory of Relativity (Einstein), 305, 392n12; frames of reference and, 271, 276, 28384; Friedmann and, 322; gravity and, 219, 27576, 28081, 291, 302, 308, 313, 318, 329, 335, 372, 395n33, 395n35, 397n44, 399nn4–5; Grossmann and, 210, 217, 258, 275, 291, 298, 303, 319, 33435; impact of, 32022; Noether and, 3045; parallelogram rule and, 31419; space and, 8083, 96; space-time and, 19697, 206, 21012; tensors and, 160, 258, 267, 274303, 323, 372n9, 389n20, 391n9, 392n12, 392n14, 394n39, 395nn32–35, 397n44; timeline for, 329, 33437; vectors and, 160

发电机12931,132,155,330

generators, 12931, 132, 155, 330

根泽尔,里恩哈德,228

Genzel, Rienhard, 228

地线,22930,28485,308,31516,392 n16,395 n32​

geodesics, 22930, 28485, 308, 31516, 392n16, 395n32

地理学(托勒密),xviii327

Geography (Ptolemy), xviii, 327

几何代数和,1,4,6,1017,346 n9,347 n13,347 nn15–16积分2122,26,31 – 34,39 – 41,350 nn7–8,352 n15 卡尔达诺和,1112,32,347 n16 克利福德238 弯曲空间和,220,22426,22931,238,383 n6 曲面219参阅曲面和,39,40;微分,109,27273,289,322,336,352 n15,393 n19 爱因斯坦,308 欧几里得和,4,74,115 17,163,210,216,220,229 – 31,303,308,327,330 四维,17,75,187,205 – 12,248,289,298,381 n18 ​汉密尔顿1,4,14,17,60-63,68,74,90,104,106,109-10,113-17,159,161,183,185,333,345 n1 向量的想法 44,58-63,68​; Klein 和,242295298381 n14;Leibniz 和,3537368 n12;Maxwell 和,11011715916118384213241289333368 n12;美索不达米亚和,xivxv;Newton 和,14394144606274308329350 n7,351 n15,368 n12;非欧几里得,117163216220284330;符号和,xxiixxiii;托勒密和,344 n6;四元数和,1036109171596318185374 n23;空间和,1474758790360 n24(参见三维空间);时空和,19019495206721013216381 n14;24142,245,259,272,28386,289,295 – 98,303 时间轴,327 30,333,336;芝诺悖论和 31,40​​

geometry: algebra and, 1, 4, 6, 1017, 346n9, 347n13, 347nn15–16; calculus and, 2122, 26, 3134, 3941, 350nn7–8, 352n15; Cardano and, 1112, 32, 347n16; Clifford and, 238; curved space and, 220, 22426, 22931, 238, 383n6; curved surfaces and, 219 (see also curved surfaces); derivatives and, 39, 40; differential, 109, 27273, 289, 322, 336, 352n15, 393n19; Einstein and, 308; Euclid and, 4, 74, 11517, 163, 210, 216, 220, 22931, 303, 308, 327, 330; four-dimensional, 17, 75, 187, 20512, 248, 289, 298, 381n18; Hamilton and, 1, 4, 14, 17, 6063, 68, 74, 90, 104, 106, 10910, 11317, 159, 161, 183, 185, 333, 345n1; ideas for vectors and, 44, 5863, 68; Klein and, 242, 295, 298, 381n14; Leibniz and, 3537, 368n12; Maxwell and, 110, 117, 159, 161, 18384, 213, 241, 289, 333, 368n12; Mesopotamia and, xivxv; Newton and, 14, 3941, 44, 60, 62, 74, 308, 329, 350n7, 351n15, 368n12; non-Euclidean, 117, 163, 216, 220, 284, 330; notation and, xxiixxiii; Ptolemy and, 344n6; quaternions and, 1036, 10917, 15963, 18185, 374n23; space and, 14, 7475, 87, 90, 360n24 (see also three-dimensional space); space-time and, 190, 19495, 2067, 21013, 216, 381n14; tensors and, 24142, 245, 259, 272, 28386, 289, 29598, 303; timeline on, 32730, 333, 336; Zeno’s paradox and, 31, 40

Gerlach,Walther ,9596,335,362 n27

Gerlach, Walther, 9596, 335, 362n27

Germain,Sophie:微积分和,25,38弯曲空间和,220,233;费马最后定理和,57;正规教育和,25,38,53 高斯,53,220 ;向量的思想和,53,57性别歧视164;时间轴,330 ;振动表面和,12627

Germain, Sophie: calculus and, 25, 38; curved space and, 220, 233; Fermat’s last theorem and, 57; formal education and, 25, 38, 53; Gauss and, 53, 220; ideas for vectors and, 53, 57; sexism and, 164; timeline on, 330; vibrating surfaces and, 12627

德文字母(麦克斯韦矢量符号),153,155

German letters (Maxwell’s symbols for vectors), 153, 155

布拉格德国大学(爱因斯坦担任教授),278

German University in Prague (Einstein as professor), 278

安德里亚·盖兹,228

Ghez, Andrea, 228

Gibbs Josiah Willard :矢量分析要素17880 Heaviside和,78,17681,184,187,196

Gibbs, Josiah Willard: Elements of Vector Analysis, 17880; Heaviside and, 78, 17681, 184, 187, 196,

吉布斯,乔赛亚·威拉德(

Gibbs, Josiah Willard (cont.)

21012333376 n12;不变性和,196;麦克斯韦和,140,143,177 79,187,212,333,377 n17 符号,78,376 n12;四元数和,177 81,184 87,376 n12,377 nn17–18 四元数和矢量代数” 185 86 时空196,210,212量和,245 时间轴333

21012, 333, 376n12; invariance and, 196; Maxwell and, 140, 143, 17779, 187, 212, 333, 377n17; notation and, 78, 376n12; quaternions and, 17781, 18487, 376n12, 377nn17–18; “Quaternions and the Algebra of Vectors,” 18586; space-time and, 196, 210, 212; tensors and, 245; timeline on, 333

吉尔·戴维(关于麦克斯韦尔),145

Gill, David (on Maxwell), 145

万向节锁,93

gimbal lock, 93

吉拉德,阿尔伯特,225

Girard, Albert, 225

Glenlair (麦克斯韦118,123,137,167,367 n9​

Glenlair (Maxwell’s home), 118, 123, 137, 167, 367n9

谷歌, 87 , 248 , 323 , 336

Google, 87, 248, 323, 336

哥廷根科学院(希尔伯特相对论方程),297

Göttingen Academy of Sciences (Hilbert’s relativity equations), 297

塞缪尔·古德斯密特96、335、362 nn27–28

Goudsmit, Samuel, 96, 335, 362nn27–28

GPS xxii203、249、279、302

GPS, x, xii, 203, 249, 279, 302

研究生麦克斯韦和,152,243,264,332;四元数/矢量和,152,376 n14 时空 211 量和,243,264

grad: Maxwell and, 152, 243, 264, 332; quaternions/vectors and, 152, 376n14; space-time and, 211; tensors and, 243, 264

Grassmann, Hermann:引理学10316163178212256331;背景,1023;Clifford 和,16163;弯曲空间和,23033;De Morgan 和,11213;微分几何和,318;Einstein 和,267;Gauss 和,1089;Gibbs 和,17881;汉密尔顿(和汉密尔顿向量)和,102 – 18,161,163,179 80,230,267,329,333,364 n3,366 n19,368 n12 拉普拉斯 105;麦克斯韦和,368 n12;莫比乌斯和,10910;报纸的11617/向量替代系统102 18,364 n3,364 n8,365 n14,366 n15,366 n19,374 n23,377 n17 吠陀和 116 时空与,196,212,215 与,245,253,267时间线上 329,331,333

Grassmann, Hermann: Ausdehnungslehre of, 10316, 163, 178, 212, 256, 331; background of, 1023; Clifford and, 16163; curved space and, 23033; De Morgan and, 11213; differential geometry and, 318; Einstein and, 267; Gauss and, 1089; Gibbs and, 17881; Hamilton (and Hamiltonian vectors) and, 10218, 161, 163, 17980, 230, 267, 329, 333, 364n3, 366n19, 368n12; Laplace and, 105; Maxwell and, 368n12; Möbius and, 10910; newspaper of, 11617; quaternion/vector alternative system and, 10218, 364n3, 364n8, 365n14, 366n15, 366n19, 374n23, 377n17; Rig Veda and, 116; space-time and, 196, 212, 215; tensors and, 245, 253, 267; timeline on, 329, 331, 333;

格拉斯曼,贾斯特斯(赫尔曼的父亲102,113

Grassmann, Justus (father of Hermann), 102, 113

格拉斯曼,贾斯特斯(赫尔曼之子)212,256

Grassmann, Justus (son of Hermann), 212, 256

格雷夫斯,约翰71,75

Graves, John, 71, 75

格雷夫斯,罗伯特71,101

Graves, Robert, 71, 101

引力:微积分20,27,34-40;弯曲空间219,237,385 n7 ;爱因斯坦305-9,313,315,318,321-22,399 nn4-5坠落,112,125,129,219,275-76,277,304,308,352 n15,353 n6,390 n5,397 n40 四维几何和,381 n18 广义相对论219,27576,280 – 81,291,302,308,313,318,329,335,372,395 n33,395 n35,397 n44,399 nn4–5 矢量的想法,46 49 ;平方反比定律和,36,125,279,315,329,367 n11,368 n15,393 n21 光和27,36 37,277 – 79,282,285,302;​​麦克斯韦和,12533142367 n11,368 n12,368 n15 ;运动和,27 34 – 36464912518221927576368 n15,397 n44;牛顿和,27344047125,130 133 , 142 , 182 , 27580 , 28587 , 3012 , 307 , 329 , 337 , 367 n11, 368 n12, 368 n15, 391 nn7–8, 397 n41;行星和,3436,133,27576,367 n11,393 n21 等效原理27678,28285和,155,182;空间92时空198 狭义相对论和,198,219,275,278 – 81,285,291,397 n40,399 n5 静止280,367 n11和,27530 3,390 n2,390 n5,391 nn7–8,393 n21,397 n41;时间轴329,335 – 37 ;轨迹,35,46 – 49,18182,274,277 波,x 20,198,302,309,313,336 – 37​​​

gravity: calculus and, 20, 27, 3440; curved space and, 219, 237, 385n7; Einstein and, 3059, 313, 315, 318, 32122, 399nn4–5; falling and, 112, 125, 129, 219, 27576, 277, 304, 308, 352n15, 353n6, 390n5, 397n40; four-dimensional geometry and, 381n18; general theory of relativity and, 219, 27576, 28081, 291, 302, 308, 313, 318, 329, 335, 372, 395n33, 395n35, 397n44, 399nn4–5; ideas for vectors and, 4649; inverse square law and, 36, 125, 279, 315, 329, 367n11, 368n15, 393n21; light and, 27, 3637, 27779, 282, 285, 302; Maxwell and, 12533, 142, 367n11, 368n12, 368n15; motion and, 27, 3436, 4649, 125, 182, 219, 27576, 368n15, 397n44; Newton and, 27, 3440, 47, 125, 130, 133, 142, 182, 27580, 28587, 3012, 307, 329, 337, 367n11, 368n12, 368n15, 391nn7–8, 397n41; planets and, 3436, 133, 27576, 367n11, 393n21; principle of equivalence and, 27678, 28285; quaternions and, 155, 182; space and, 92; space-time and, 198; special theory of relativity and, 198, 219, 275, 27881, 285, 291, 397n40, 399n5; stationary, 280, 367n11; tensors and, 275303, 390n2, 390n5, 391nn7–8, 393n21, 397n41; timeline for, 329, 33537; trajectories and, 35, 4649, 18182, 274, 277; as waves, x, 20, 198, 302, 309, 313, 33637

重力探测器B卫星,336

Gravity Probe B satellite, 336

希腊数学家,xiv、xxvi;代数和,5、7、9、13 天文学 xvii 微积分和,22、29;弯曲空间和,23334向量的思想和,49、64影响xvi xvii ;麦克斯韦153;穷竭法和,2226 纳布拉和,143;奥斯曼人和,47 ;时空和,90 ;泰特和,143、147、178 参阅特定个人

Greek mathematicians, xiv, xxvi; algebra and, 5, 7, 9, 13; astronomy and, xvii; calculus and, 22, 29; curved space and, 23334; ideas for vectors and, 49, 64; influence of, xvixvii; Maxwell and, 153; method of exhaustion and, 2226; nabla and, 143; Ottomans and, 47; space-time and, 90; Tait and, 143, 147, 178. See also specific individuals

格林,乔治126,131

Green, George, 126, 131

Grossmann, Marcel:弯曲空间和,2172023023839;爱因斯坦和,2171923839244253258275283285952989930331933334Entwurf理论和,29095393 n21;广义相对论和,21021725827529129830331933435;Ricci和,219,238,244,253,258,275,285 – 87,290,295,303,319,334 ;量和 244,253,258,275,282 95,298 99,303 时间轴开启 333 – 35​​

Grossmann, Marcel: curved space and, 21720, 230, 23839; Einstein and, 21719, 23839, 244, 253, 258, 275, 283, 28595, 29899, 303, 319, 33334; Entwurf theory and, 29095, 393n21; general theory of relativity and, 210, 217, 258, 275, 291, 298, 303, 319, 33435; Ricci and, 219, 238, 244, 253, 258, 275, 28587, 290, 295, 303, 319, 334; tensors and, 244, 253, 258, 275, 28295, 29899, 303; timeline on, 33335

论,204,208,362 n30​

group theory, 204, 208, 362n30

格鲁纳特,约翰,110

Grunert, Johann, 110

火药阴谋,50

Gunpowder Plot, 50

资格论文,232,237-38,257,312-13

habilitation theses, 232, 23738, 257, 31213

哈布斯堡王朝,47

Habsburgs, 47

哈雷,埃德蒙,37,351 n13

Halley, Edmond, 37, 351n13

汉密尔顿,阿奇博尔德,69岁

Hamilton, Archibald, 69

汉密尔顿,海伦,127375357 n7

Hamilton, Helen, 12, 73, 75, 357n7

汉密尔顿,海伦·伊丽莎,73岁

Hamilton, Helen Eliza, 73

汉密尔顿,詹姆斯,6465

Hamilton, James, 6465

汉密尔顿,威廉·埃德温,69岁

Hamilton, William Edwin, 69

汉密尔顿,威廉·罗文:代数和,18141754576063697174768083868990991071161511611631787918322924333233345 n1,345 n9,346 n2,358 n9 算术,xxvxxvi;阿姆斯特朗和,3 天文学和1,7,62,65,73,76,102;背景,6468 布鲁姆桥和,1 3,4142,75,78,97,145,243,325,331 布鲁姆涂鸦含义23,76,359 n13 ;微积分1820,41,65;凯莱和,8,8183,86,90,137,179,332硬币矢量xixii交换律和,2,61,76,79,90,99,104,107,11416,163,178,332,345 n9,359 n13 复数,54​5760646769747578100104107109114116178244355 n20、356 n3、358 n11;夫妻,61 – 64 、67 、71 、355 nn20–21 晶体20弯曲空间 229 30 死亡,1464720,41,63,152,298;爱因斯坦和,306 – 7,310,315 ;方程的优雅,325 ;四维数学2,4,17,7578,91,98 99,206 ;天才147 几何 1,4,14,17,60 – 63,68,74,90,104,106,10910,113 17,159,161,183,185,206,333,345 n1 Grassmann 和, 10218 , 161 , 163 , 17980 , 230 , 267 , 329 , 333 , 364 n3, 366 n19, 368 n12; 向量的想法和, 42 , 52 , 54 , 57 , 6068 , 355 nn20–21; 虚数和, 2 , 6 , 52 , 57 , 60 , 7478 , 153 , 171 , 358 n11; 语言, 64 ; 拉普拉斯和, 65 , 104 ;模量定律7172、7576、356 n3、358 n11元数讲座76、107、114、117、119、142 – 43、260n24, 345 n1, 355 n20, 357 n4; 光和, 2 , 1920 ; 疯帽子戏法猜想和, 345 n9; 麦克斯韦和, 11819 , 128 , 13745 , 368 n12; nabla 和, 14344 , 146 , 148 , 15253 , 243 , 298 , 332 ;牛顿42,52,60,62 - 65,74,84,103,146 - 47,181,196,306,325,355 n20,358 n12,367 n12 符号和,373 n12 ;月全食月亮 98 ;诗歌和62 - 64,73,75,84,98,331 作为神童64 - 68 ;​​​​四元数和, 3 , 78 , 14 , 17 , 6465 , 71 , 7483 , 86 , 9091 , 97119 , 128 , 14253 , 15963 , 171 , 17886 , 196 , 206 , 229 , 24344 , 325 , 33133 , 336 , 355 n20, 359 n14, 360 n24, 364 n3, 366 nn18–19, 372 n10; 折射和, 18 , 20 ,331;声誉,73;利玛窦和,244267298325;旋转和,131454606768899197100104107114178183336360 n24;破坏规则和,xxvixxvii8081;空间和,146886899197100106711416162183356 n3,357 n7,358 n11,359 nn13–14,360 n24;时空和,196 ,206 ,381 n17量和 243 45 267 298 395 n34;时间轴,329 – 33 ,336 都柏林学院和,6566;三元组,68717576

Hamilton, William Rowan: algebra and, 18, 14, 17, 54, 57, 6063, 6971, 74, 76, 8083, 86, 8990, 99107, 116, 151, 161, 163, 17879, 183, 229, 243, 33233, 345n1, 345n9, 346n2, 358n9; arithmetic of, xxvxxvi; Armstrong and, 3; astronomy and, 1, 7, 62, 65, 73, 76, 102; background of, 6468; Broome Bridge and, 13, 4142, 75, 78, 97, 145, 243, 325, 331; Broome Bridge graffiti, meaning of, 23, 76, 359n13; calculus and, 1820, 41, 65; Cayley and, 8, 8183, 86, 90, 137, 179, 332; coins vector term, xixii; commutative law and, 2, 61, 76, 79, 90, 99, 104, 107, 11416, 163, 178, 332, 345n9, 359n13; complex numbers and, 54, 57, 6064, 67, 69, 7475, 78, 100, 104, 107, 109, 114, 116, 178, 244, 355n20, 356n3, 358n11; couples of, 6164, 67, 71, 355nn20–21; crystals and, 20; curved space and, 22930; death of, 14647; derivatives and, 20, 41, 63, 152, 298; Einstein and, 3067, 310, 315; elegance of equation of, 325; four-dimensional mathematics and, 2, 4, 17, 7578, 91, 9899, 206; genius of, 147; geometry and, 1, 4, 14, 17, 6063, 68, 74, 90, 104, 106, 10910, 11317, 159, 161, 183, 185, 206, 333, 345n1; Grassmann and, 10218, 161, 163, 17980, 230, 267, 329, 333, 364n3, 366n19, 368n12; ideas for vectors and, 42, 52, 54, 57, 6068, 355nn20–21; imaginary numbers and, 2, 6, 52, 57, 60, 7478, 153, 171, 358n11; languages of, 64; Laplace and, 65, 104; law of moduli of, 7172, 7576, 356n3, 358n11; Lectures on Quaternions, 76, 107, 114, 117, 119, 14243, 260n24, 345n1, 355n20, 357n4; light and, 2, 1920; Mad Hatter parody conjecture and, 345n9; Maxwell and, 11819, 128, 13745, 368n12; nabla and, 14344, 146, 148, 15253, 243, 298, 332; Newton and, 42, 52, 60, 6265, 74, 84, 103, 14647, 181, 196, 306, 325, 355n20, 358n12, 367n12; notation and, 373n12; Ode to the Moon under Total Eclipse, 98; poetry and, 6264, 73, 75, 84, 98, 331; as prodigy, 6468; quaternions and, 3, 78, 14, 17, 6465, 71, 7483, 86, 9091, 97119, 128, 14253, 15963, 171, 17886, 196, 206, 229, 24344, 325, 33133, 336, 355n20, 359n14, 360n24, 364n3, 366nn18–19, 372n10; refraction and, 18, 20, 331; reputation of, 73; Ricci and, 244, 267, 298, 325; rotation and, 1, 3, 14, 54, 60, 6768, 8991, 97, 100, 104, 107, 114, 178, 183, 336, 360n24; rule-breaking and, xxvixxvii, 8081; space and, 14, 6886, 8991, 97100, 1067, 114, 16162, 183, 356n3, 357n7, 358n11, 359nn13–14, 360n24; space-time and, 196, 206, 381n17; tensors and, 24345, 267, 298, 395n34; timeline on, 32933, 336; Trinity College Dublin and, 6566; triples of, 6871, 7576

汉密尔顿日,12

Hamilton Day, 12

汉密尔顿过程,355 , 355 n20

Hamilton’s process, 355, 355n20

Harriot, Thomas:代数和,89141729325261112328348 nn18–19,349 n22,350 n7,353 n5;弹道学(抛射轨迹)和,49277353 n6 ;微积分和,27293132350 nn7–8;弯曲空间和,22122428384 n6;死亡,256;笛卡尔和,32350 n8;下落运动和,49112 ;火药阴谋和,50;向量概念4852,61,328,353 nn5–8,354 n9,354 n12 无穷级数和,31 拉格朗日和348 n18 碰撞力学 5052运动和,4849,112,353 n6,353 n8 ;Praxis8,32,52,328,347 n10,349 n22,350 n8;塞尔特曼和16 – 17 符号主义8,1417,29,11112,349 nn21–22 ​对称性和,353 n8 ;张量和,256,277 Thomas Harriot:《科学人生》347 n10;时间轴,328;Viète 和,15;Wallis 和,1415 , 27 , 3132 , 52 , 61 , 111 , 348 n18, 350 n7, 354 n10

Harriot, Thomas: algebra and, 89, 1417, 2932, 52, 61, 112, 328, 348nn18–19, 349n22, 350n7, 353n5; ballistics (projectile trajectory) and, 49, 277, 353n6; calculus and, 27, 29, 3132, 350nn7–8; curved space and, 221, 22428, 384n6; death of, 256; Descartes and, 32, 350n8; falling motion and, 49, 112; Gunpowder Plot and, 50; ideas for vectors and, 4852, 61, 328, 353nn5–8, 354n9, 354n12; infinite series and, 31; Lagrange and, 348n18; mechanics of collisions and, 5052; motion and, 4849, 112, 353n6, 353n8; Praxis, 8, 32, 52, 328, 347n10, 349n22, 350n8; Seltman and, 1617; symbolism and, 8, 1417, 29, 11112, 349nn21–22; symmetry and, 353n8; tensors and, 256, 277; Thomas Harriot: A Life in Science, 347n10; timeline on, 328; Viète and, 15; Wallis and, 1415, 27, 3132, 52, 61, 111, 348n18, 350n7, 354n10

霍金,斯蒂芬228,385 n7

Hawking, Stephen, 228, 385n7

海瑟姆,阿布·阿里·伊本·阿勒,27 岁

Haytham, Abu Ali Ibn al-, 27

HeavisideOliver:背景16970 ; 电磁理论矢量分析),170,176 – 77 ;吉布斯和,78,17681,184,187,196,21012,333,376 n12;不变性196 ;麦克斯韦(和麦克斯韦方程16978,187,212,333,369 n21,372 nn2–10,375 n10,376 nn11–13,377 n15 现代矢量符号78,143,17273 “论电流的能量”,170;四元数与矢量和,153 169 81 184 87 375 nn2–10,376 nn11–13,377 n15;时空和,19620621012;电报和,16972177;时间线,333

Heaviside, Oliver: background of, 16970; Electromagnetic Theory (and vector analysis), 170, 17677; Gibbs and, 78, 17681, 184, 187, 196, 21012, 333, 376n12; invariance and, 196; Maxwell (and Maxwell’s equations) and, 16978, 187, 212, 333, 369n21, 372nn2–10, 375n10, 376nn11–13, 377n15; modern vector notation of, 78, 143, 17273; “On the Energy of Electric Currents,” 170; quaternions vs. vectors and, 153, 16981, 18487, 375nn2–10, 376nn11–13, 377n15; space-time and, 196, 206, 21012; telegraphy and, 16972, 177; timeline of, 333

亨利五世(莎士比亚),49

Henry V (Shakespeare), 49

赫歇尔,卡罗琳,6667

Herschel, Caroline, 6667

赫歇尔,约翰,62 , 6667 , 1012 , 108

Herschel, John, 62, 6667, 1012, 108

赫歇尔·威廉(Herschel, William)67 岁

Herschel, William, 67

赫兹,海因里希逝世,175 麦克斯韦和,141,160,377 n15;无线电波和,141,159,168,241 时间轴333矢量和,175

Hertz, Heinrich: death of, 175; Maxwell and, 141, 160, 377n15; radio waves and, 141, 159, 168, 241; timeline on, 333; vectors and, 175

Hestenes,David,374 n23

Hestenes, David, 374n23

希格斯色子,20,99,320

Higgs boson, 20, 99, 320

希尔伯特,戴维:比安奇恒等式和,310,395 n35 “致文化世界宣言”和,293爱因斯坦29398,301,304 – 6,310,313,317 – 18,335,395 nn33–35,396 nn36–37,397 n43;“物理学基础297;广义相对论包括优先权辩论)和29598,301;克莱和,293,295,298,3046,310 – 13,335 列维奇维塔,294 98;Mie 和,296;Minkowski 和,211293;Noether 和,304631013335;张量和,29498301;时间轴,335

Hilbert, David: Bianchi identities and, 310, 395n35; “Declaration to the Cultural World” and, 293; Einstein and, 29398, 301, 3046, 310, 313, 31718, 335, 395nn33–35, 396nn36–37, 397n43; “Foundations of Physics,” 297; general relativity (incl. priority debate) and, 29598, 301; Klein and, 293, 295, 298, 3046, 31013, 335; Levi-Civita and, 29498; Mie and, 296; Minkowski and, 211, 293; Noether and, 3046, 31013, 335; tensors and, 29498, 301; timeline on, 335

Hinton,Charles Howard,4 - D几何和,206,381 n16

Hinton, Charles Howard, 4-D geometry and, 206, 381n16

霍布斯,托马斯,34 , 39

Hobbes, Thomas, 34, 39

霍奇,威廉,316

Hodge, William, 316

胡克,罗伯特,3637351 nn12–13

Hooke, Robert, 3637, 351nn12–13

匈牙利,47

Hungary, 47

惠更斯,克里斯蒂安20,112

Huygens, Christian, 20, 112

希帕提娅xvii7,328

Hypatia, xvii, 7, 328

超立方体,206

hypercubes, 206

图像压缩,8690

image compression, 8690

代数和,23,6,1216,347 n10;微积分和,21笛卡尔和,6,15,52,346 n8;欧拉和,52 – 57汉密尔顿和,2,6,52,57,60,74 78,153,171,358 n11 向量思想5259,68麦克斯韦和,143 夫斯基和,381 n20 乘法作为旋转,59 数和,153,171,187 空间和,6970,74 78 ;时间轴328

imaginary numbers: algebra and, 23, 6, 1216, 347n10; calculus and, 21; Descartes and, 6, 15, 52, 346n8; Euler and, 5257; Hamilton and, 2, 6, 52, 57, 60, 7478, 153, 171, 358n11; ideas for vectors and, 5259, 68; Maxwell and, 143; Minkowski and, 381n20; multiplication as a rotation of, 59; quaternions and, 153, 171, 187; space and, 6970, 7478; timeline on, 328

印加数学,xvii

Inca mathematics, xvii

索引符号:弯曲空间和,235 36 现代观点,26465;张量和,25354,258 – 71

index notation: curved space and, 23536; modern view of, 26465; tensors and, 25354, 25871

印度数学xvii5-6,9,82,328

Indian mathematics, xvii, 56, 9, 82, 328

感应),133,138,151,155,369 n21,375 n10​

induction (magnetic), 133, 138, 151, 155, 369n21, 375n10

工业革命,83

Industrial Revolution, 83

惯性系,277,278,283-84,397n40

inertial frame, 277, 278, 28384, 397n40

无穷小:弯曲空间和,22124383 n6;发展,2130343940;霍布斯和,34;积分微积分和,213034394012222124350 n6,383 n6;莱布尼茨和,2630350 n6;穷举法和,2226;牛顿和,263030;芝诺和,2630

infinitesimals: curved space and, 22124, 383n6; development of, 2130, 34, 39, 40; Hobbes and, 34; integral calculus and, 2130, 34, 39, 40, 122, 22124, 350n6, 383n6; Leibniz and, 2630, 350n6; method of exhaustion and, 2226; Newton and, 2630, 30; Zeno and, 2630

无穷大,21,29,32,53,350 n6​​

infinity, 21, 29, 32, 53, 350n6

积分积分):代数和,22近似面积和,21 边界 132 – 33 弯曲空间和,225,383 n6 发展,2027,31,35;爱因斯坦3056 无穷小21 – 30,34,39,40,122,221 – 24,350 n6,383 n6 线122,138,151,256,368 n12,369 n21,373 n11 Maxwell和,1223313740367 n8,368 n12,368 n15,369 n19,369 n21;穷尽法和,22 – 26 表面122 128 130 ,1331373815222526256367 n8,369 n19,369 n21,373 n11时间轴开启,328

integrals (integral calculus): algebra and, 22; approximating areas and, 21; boundaries of, 13233; curved space and, 225, 383n6; development of, 2027, 31, 35; Einstein and, 3056; infinitesimals and, 2130, 34, 39, 40, 122, 22124, 350n6, 383n6; line, 122, 138, 151, 256, 368n12, 369n21, 373n11; Maxwell and, 12233, 13740, 367n8, 368n12, 368n15, 369n19, 369n21; method of exhaustion and, 2226; surface, 122, 128, 130, 133, 13738, 152, 22526, 256, 367n8, 369n19, 369n21, 373n11; timeline on, 328

干涉图案,19,97,199,363n31

interference patterns, 19, 97, 199, 363n31

国际天文学联合会,xviii337

International Astronomical Union, xviii, 337

国际数学联合会,317

International Mathematics Union, 317

不变性:美丽的概念,18997;凯莱和,19194,255;坐标变换和,19094,2024,237,256,265,26869,305,307,389 n22,390 n23弯曲空间226 30,235 38 爱因斯坦和 305 8,323,399 n5 高斯和,22630 吉布斯和,196 海维赛德和,196;诺特和,3068,399 n5 Ricci 和,24244 , 25659 , 26263 , 26973 , 285 , 290 , 298 ; 时空和,18997 , 2025 , 208 , 212 , 216 , 378 n4, 381 n14; Tait 和,19194 ; 张量和, 24244 , 25559 , 26269 , 27273 , 285 , 290 , 389 n22, 390 n23; Thomson 和,194 ; 拓扑和,22630

invariance: beautiful concept of, 18997; Cayley and, 19194, 255; coordinate transformations and, 19094, 2024, 237, 256, 265, 26869, 305, 307, 389n22, 390n23; curved space and, 22630, 23538; Einstein and, 3058, 323, 399n5; Gauss and, 22630; Gibbs and, 196; Heaviside and, 196; Noether and, 3068, 399n5; Ricci and, 24244, 25659, 26263, 26973, 285, 290, 298; space-time and, 18997, 2025, 208, 212, 216, 378n4, 381n14; Tait and, 19194; tensors and, 24244, 25559, 26269, 27273, 285, 290, 389n22, 390n23; Thomson and, 194; topology and, 22630

“不变变分问题” (Noether),306

“Invariante Variationsprobleme” (Noether), 306

平方反比定律:代数34,36,125,138,154,277,279,315,329,367 n11,368 n15,393 n21 ;黑洞,315 ;积分,34,36 引力和,36,125,279,315,329,367 n11,368 n15,393 n21;麦克斯韦和125,138,367 n11,368 n15 ​​牛顿和,34,36,125,277,279,329,367 n11 ; 154 和,277,279,393 n21 ;时间轴开启329

inverse square law: algebra and, 34, 36, 125, 138, 154, 277, 279, 315, 329, 367n11, 368n15, 393n21; black holes and, 315; calculus and, 34, 36; gravity and, 36, 125, 279, 315, 329, 367n11, 368n15, 393n21; Maxwell and, 125, 138, 367n11, 368n15; Newton and, 34, 36, 125, 277, 279, 329, 367n11; quaternions and, 154; tensors and, 277, 279, 393n21; timeline on, 329

伊拉克,xiii10

Iraq, xiii, 10

铁屑,95 , 13440 , 174 , 183

iron filings, 95, 13440, 174, 183

伊斯兰数学5,11,47

Islamic mathematics, 5, 11, 47

詹姆斯一世,50岁

James I, 50

詹姆斯·韦伯太空望远镜,192

James Webb Space Telescope, 192

杰弗里斯·哈罗德,321

Jeffreys, Harold, 321

犹太人85,196,317

Jews, 85, 196, 317

琼斯,威廉,5253

Jones, William, 5253

焦耳秒, 81

joule-seconds, 81

木星,199

Jupiter, 199

康德,伊曼纽尔62,64

Kant, Immanuel, 62, 64

基思奖章(泰特获胜),159

Keith Medal (Tait wins), 159

开尔文温标,148

Kelvin temperature scale, 148

Khayyam,奥马尔,347 n15

Khayyam, Omar, 347n15

穆罕默德·本·穆萨·花剌子米,58 , 1011 , 13 , 328 , 346 n9

Khwārizmī, Mohammed ibn Mūsā al-, 58, 1011, 13, 328, 346n9

动能,9,352 nn14–15,353 n8,399 n4​​​

kinetic energy, 9, 352nn14–15, 353n8, 399n4

克莱因,费利克斯:奇泽姆和,242;“致文化世界宣言”和,293;爱因斯坦和,304631013;几何和,242295298381 n14;希尔伯特和,293295298304631013335数学年鉴和,25556;诺特和,304631031213335381 n14;符号和,222;里奇和,242256274334;时空与212;张242、25556、27475、293 – 95、298 ;时间轴334 35

Klein, Felix: Chisholm and, 242; “Declaration to the Cultural World” and, 293; Einstein and, 3046, 31013; geometry and, 242, 295, 298, 381n14; Hilbert and, 293, 295, 298, 3046, 31013, 335; Mathematische Annalen and, 25556; Noether and, 3046, 310, 31213, 335, 381n14; notation and, 222; Ricci and, 242, 256, 274, 334; space-time and, 212; tensors and, 242, 25556, 27475, 29395, 298; timeline on, 33435

克莱,托尼,97,336,363 n31

Klein, Tony, 97, 336, 363n31

科瓦列夫斯基,索尼娅,159

Kovalevsky, Sonia, 159

拉格朗日,约瑟夫-路易斯: 分析力学1045,329法和,305;度量衡委员会,105,329,364 n6;爱因斯坦和,279,305 – 7;电和,125 – 26,130;格拉斯曼和,104 – 5 哈里奥特和,348 n18 麦克斯韦,125 30,368 n12牛顿引力125,279牛顿运动 105,307 理论和,12526,368 n12;时间轴,329 矢量128

Lagrange, Joseph-Louis: Analytical Mechanics, 1045, 329; calculus of variations and, 305; Committee on Weights and Measures, 105, 329, 364n6; Einstein and, 279, 3057; electricity and, 12526, 130; Grassmann and, 1045; Harriot and, 348n18; Maxwell and, 12530, 368n12; Newtonian gravity and, 125, 279; Newtonian motion and, 105, 307; potential theory and, 12526, 368n12; timeline on, 329; vectors and, 128

拉普拉斯,皮埃尔-西蒙: 分析力学 104 5,329 度量衡委员会,105,329 电力和,127,130 ;格拉斯曼和,105 汉密尔顿和,65,104麦克斯韦126 – 30,144,369 n21 理论和,126;时间轴329 天体力学论述65,104,105,126,329

Laplace, Pierre-Simon: Analytical Mechanics, 1045, 329; Committee on Weights and Measures, 105, 329; electricity and, 127, 130; Grassmann and, 105; Hamilton and, 65, 104; Maxwell and, 12630, 144, 369n21; potential theory and, 126; timeline on, 329; Treatise on Celestial Mechanics, 65, 104, 105, 126, 329

拉普拉斯算子(运算符),12728 , 137 , 142 , 144 , 150

Laplacian (operator), 12728, 137, 142, 144, 150

大型强子对撞机,320

Large Hadron Collider, 320

大型语言模型(LLM) ,24950,387 n12,388 n13

large language models (LLMs), 24950, 387n12, 388n13

激光干涉引力波天文台(LIGO),198,336-37

Laser Interferometer Gravitational-wave Observatory (LIGO), 198, 33637

模数定律, 7172 , 7576 , 356 n3, 358 n11

law of moduli, 7172, 7576, 356n3, 358n11

运动定律:微积分和,3435;弯曲空间和,230;爱因斯坦和,306;矢量的概念和,434448353 n8;麦克斯韦和,125;牛顿和,343643444849125182202271275283306358 n12,368 n15 空间和,358 n12;张量和,271

laws of motion: calculus and, 3435; curved space and, 230; Einstein and, 306; ideas for vectors and, 4344, 48, 353n8; Maxwell and, 125; Newton and, 3436, 4344, 4849, 125, 182, 202, 271, 275, 283, 306, 358n12, 368n15; space and, 358n12; tensors and, 271

讲座和论文(Clifford),166

Lectures and Essays (Clifford), 166

四元数讲座(汉密尔顿),117,345 n1;格拉斯曼和,107,114 向量的思想和,355 n20 麦克斯韦和,119,142 – 43 ;空间和,76,357 n4,360 n24 ;出版麻烦 106 7

Lectures on Quaternions (Hamilton), 117, 345n1; Grassmann and, 107, 114; ideas for vectors and, 355n20; Maxwell and, 119, 14243; space and, 76, 357n4, 360n24; trouble in publishing, 1067

莱布尼茨,戈特弗里德·威廉,2;背景,28 ;微积分27303541349 n5,350 n6;对牛顿理论的批判和,35;弯曲空间和,223;导数和,39368 n12;动力学论文4445;几何和,3537368 n12;汉密尔顿和,65;矢量思想和,4445546265;无穷小和,263030350 n6;符号39,40,41,65,105,111-12,368 n12牛顿优先权争议41,44,62,72/矢量和,111-12,366 n15 空间 62,72,83 ;时空和,196 时间轴 328-30​

Leibniz, Gottfried Wilhelm, 2; background of, 28; calculus and, 2730, 3541, 349n5, 350n6; criticism of Newton’s theory and, 35; curved space and, 223; derivatives and, 39, 368n12; Essay on Dynamics, 4445; geometry and, 3537, 368n12; Hamilton and, 65; ideas for vectors and, 4445, 54, 62, 65; infinitesimals and, 2630, 30, 350n6; notation and, 39, 40, 41, 65, 105, 11112, 368n12; priority dispute with Newton, 41, 44, 62, 72; quaternions/vectors and, 11112, 366n15; space and, 62, 72, 83; space-time and, 196; timeline on, 32830

轻子,99

leptons, 99

列维-奇维塔,利比里亚特雷维萨尼,317

Levi-Civita, Liberia Trevisani, 317

Levi - Civita Tullio 背景27475弯曲空间和21920,237;爱因斯坦和,219,310,31519希尔伯特29498 里奇21920,274,285,303,316,319,334和,274,285,289,294 – 98,303 ;时间轴334 35

Levi-Civita, Tullio: background of, 27475; curved space and, 21920, 237; Einstein and, 219, 310, 31519; Hilbert and, 29498; Ricci and, 21920, 274, 285, 303, 316, 319, 334; tensors and, 274, 285, 289, 29498, 303; timeline on, 33435

刘易斯,乔治,16365

Lewes, George, 16365

莱顿瓶,124

Leyden jar, 124

:黑洞和,228,302,313,315,337,385 n7,397 n44,401 n12衍射19,277 – 78,293,302 – 3,317,335,337,397 n44 日食98,199,294,3023,317,320,335,397 n44 ​​​​爱因斯坦和, 19 , 35 , 141 , 20020 1, 205 , 212 , 27785 , 29394 , 303 , 317 , 335 , 337 , 347 n12, 379 n10, 397 n44; 电, 129 ; 引力和, 27 , 3637 , 27779 , 282 , 285 , 302 ; 汉密尔顿和, 2 , 1920 ; 惠更斯和, 112 ; 干涉图案和, 19 , 97 , 199 , 363 n31;麦克斯韦和,1618351414217319820 0,212 ,332 349 n1,387 n10;迈克尔逊–莫雷实验和,19899380 n11;性质,16 18 20 牛顿和,18193637202277280285 303,337 397 n44; 光学和, 18 , 20 , 27 , 123 , 147 ; 作为粒子, 1819 , 37 , 97 , 277 ; 光子和, 1920 , 99 , 277 , 284 , 348 n17, 361 n26, 362 n30; 量子理论和, 19 , 199 , 347 ; 红移和, 27881 , 33637 ; 折射, 18 , 20 , 66 , 102 , 331 ; 光谱, 96 , 336 , 361 n26;速度, 199,202-3,205,212,264,278-85,376n12,379n10,382n24,389n20 ;实验和, 97,141 ;, 16-20,97-98,112,141,173,198-99,278,302,332,337,349n1,361n26,376n11 ;, 18-19,97,141,199,330

light: black holes and, 228, 302, 313, 315, 337, 385n7, 397n44, 401n12; diffraction of, 19, 27778, 293, 3023, 317, 335, 337, 397n44; eclipses, 98, 199, 294, 3023, 317, 320, 335, 397n44; Einstein and, 19, 35, 141, 200201, 205, 212, 27785, 29394, 303, 317, 335, 337, 347n12, 379n10, 397n44; electric, 129; gravity and, 27, 3637, 27779, 282, 285, 302; Hamilton and, 2, 1920; Huygens and, 112; interference patterns and, 19, 97, 199, 363n31; Maxwell and, 16, 18, 35, 14142, 173, 198200, 212, 332, 349n1, 387n10; Michelson-Morley experiment and, 19899, 380n11; nature of, 16, 1820; Newton and, 1819, 3637, 202, 277, 280, 285, 303, 337, 397n44; optics and, 18, 20, 27, 123, 147; as particle, 1819, 37, 97, 277; photons and, 1920, 99, 277, 284, 348n17, 361n26, 362n30; quantum theory and, 19, 199, 347; redshift and, 27881, 33637; refraction of, 18, 20, 66, 102, 331; spectra of, 96, 336, 361n26; speed of, 199, 2023, 205, 212, 264, 27885, 376n12, 379n10, 382n24, 389n20; two-slit experiment and, 97, 141; as wave, 1620, 9798, 112, 141, 173, 19899, 278, 302, 332, 337, 349n1, 361n26, 376n11; Young and, 1819, 97, 141, 199, 330

“写下这些文字时,我坚信在十一月火熄灭后再读数学是不明智的!”(麦克斯韦),121

“Lines Written under the Conviction That It Is Not Wise to Read Mathematics in November after One’s Fire Is Out!” (Maxwell), 121

列表约翰,111,227

Listing, Johann, 111, 227

劳埃德·汉弗莱20岁

Lloyd, Humphrey, 20

罗巴切夫斯基,尼古拉11516,220

Lobachevsky, Nicolai, 11516, 220

洛奇,奥利弗168,171

Lodge, Oliver, 168, 171

伦敦数学学会,151

London Mathematical Society, 151

伦敦皇家学会,157

London’s Royal Institution, 157

伦敦时报(宣布证实爱因斯坦的光弯曲预测),303

London Times (announcing confirmation of Einstein’s light bending prediction), 303

洛伦兹,亨德里克,96;弯曲空间和,227;爱因斯坦和,2001224228028384334380 n12,381 n14,392 n12,394 n30,399 n5;电子理论和,380 n12;以太和,199 205210280;麦克斯韦定律和,175 迈克尔逊– 莫雷实验和,199 207庞加莱和,19920 1,207,280 时空与,19921 2,380 n12,381 n14,381 n20,382 n24;自旋96,362 n28;张量,242 43,262,269,27980,28384,288,292,300,392 n18 时间轴,334 ​​​变换19920 9,227,242 – 43,262,269,279,28384,288,300,381 n20,382 n24,392 n18,399 n5 ;向量和 207​​

Lorentz, Hendrik, 96; curved space and, 227; Einstein and, 20012, 242, 280, 28384, 334, 380n12, 381n14, 392n12, 394n30, 399n5; electron theory and, 380n12; ether and, 199, 205, 210, 280; Maxwell’s laws and, 175; Michelson-Morley experiment and, 199, 207; Poincaré and, 199201, 207, 280; space-time and, 199212, 380n12, 381n14, 381n20, 382n24; spin and, 96, 362n28; tensors and, 24243, 262, 269, 27980, 28384, 288, 292, 300, 392n18; timeline on, 334; transformations of, 199209, 227, 24243, 262, 269, 279, 28384, 288, 300, 381n20, 382n24, 392n18, 399n5; vectors and, 207

洛夫莱斯,艾达,7374149

Lovelace, Ada, 7374, 149

伦托尼, 155 , 372 n9, 398 n44

Lun, Tony, 155, 372n9, 398n44

麦克法兰,亚历山大 181,181-82

Macfarlane, Alexander, 181, 18182

机器学习:人工智能(AI)和,82,8690,221,24748,32223;分类和,88扩展86 90 矩阵和,86 90预测和,88 ;张量和,88,247 ;向量和,8688

machine learning: artificial intelligence (AI) and, 82, 8690, 221, 24748, 32223; classifications and, 88; expansion of, 8690; matrices and, 8690; predictions and, 88; tensors and, 88, 247; vectors and, 8688

疯帽子戏仿猜想,xxvi2,345 n9

Mad Hatter parody conjecture, xxvi, 2, 345n9

磁场矢量麦克斯韦和,118,155,201-2,264,287 392n18​

magnetic field vector: Maxwell and, 118, 155, 2012, 264, 287; tensors and, 392n18

磁共振成像(MRI xxv9798,321

magnetic resonance imaging (MRI), xxv, 9798, 321

马可尼,古列莫,168

Marconi, Gugliemo, 168

马里奇米列娃,19720 0、293、333、335、379 nn9–10、394 n31

Marić, Mileva, 197200, 293, 333, 335, 379nn9–10, 394n31

马歇尔学院(麦克斯韦教授),137 , 369 n18

Marischal College (Maxwell a professor at), 137, 369n18

马洛,克里斯托弗,49岁

Marlowe, Christopher, 49

质量密度,287,391 n8

mass density, 287, 391n8

数学年鉴(期刊),25556

Mathematische Annalen (journal), 25556

矩阵:凯莱矩阵8286,8990,230,332 ;弯曲空间和,230,384 n6 ;狄拉克矩阵和,96,320,323,336,362 n29 因式分解,86 万向锁和,93 ;图像压缩和,8690 ;线性方程和,83,85,89,246,260,324 ;机器学习8690 泡利自旋,336,362 n30 元数和,162,362 n29 机器人和,8690 搜索引擎和,8690;空间71,8193,96,360 n24,361 n25,362 nn29–30;张量24550,253 – 55,258 – 65,270,324,388 n19,389 n22,390 n23,392 n18 时间轴开启 332,336 变换246,259 61,263,360 n24,388 n19,390 n23,392 n18​​​

matrices: Cayley and, 8286, 8990, 230, 332; curved space and, 230, 384n6; Dirac and, 96, 320, 323, 336, 362n29; factorisation of, 86; gimbal lock and, 93; image compression and, 8690; linear equations and, 83, 85, 89, 246, 260, 324; machine learning and, 8690; Pauli spin, 336, 362n30; quaternions and, 162, 362n29; robots and, 8690; search engines and, 8690; space and, 71, 8193, 96, 360n24, 361n25, 362nn29–30; tensors and, 24550, 25355, 25865, 270, 324, 388n19, 389n22, 390n23, 392n18; timeline on, 332, 336; transformation, 246, 25961, 263, 360n24, 388n19, 390n23, 392n18

皮埃尔·路易斯·莫罗·莫佩尔蒂,3839

Maupertuis, Pierre-Louis Moreau, 3839

麦克斯韦,詹姆斯克拉克,xi;抽象概念和, 118;超距作用与场和,13237;代数和,3716;类比和,129301321353615152;动物和,120158160;阿基米德和,122;天文学和,145;背景,11819;微积分和,1835349 nn1–2;坎贝尔和,11915015315716667;笛卡尔坐标和,128143;凯莱和,137颜色123、158、248库仑定律138,154,367 n11,370 n21 ;收敛153,155,161,172,372 n7 ;弯曲空间和,219,231,23539;消亡16667;德·摩根145,126 – 27,137 – 38,141,152,172,264,271,300,369 n21,376 n14 积分127,13741,369 n19,369 n21发散和,144,149,15255,161,172,243,264,332,369 n21,372 n7,375 n10 爱因斯坦 xii 7,35,129,141,160,183,196,200 20 1,205,207,212,238,241 42,280,287,289,325,334,366 n1,380 n12 ;​​​电学和,12345,150,154,157,168,332,343 n2,349 n2,365 n14,367 n11,368 n14,369 n19,369 n21,372 n7,376 n14 电磁3,16,110,118 46 ​​​​​15051 , 15660 , 17073 , 177 , 183 , 19820 4 , 213 , 241 , 271 , 280 , 28890 , 320 , 325 , 33233 , 371 n23, 376 n11; 电动势强度和, 153 ; 电子和, 12425 , 129 ; 欧拉和, 136 ; 法拉第和, 12829 , 13342 , 149 , 151 , 167 , 237 , 279 , 289 , 331 , 369 n21;通量和,1303313738369 nn20–21;高斯和,12613013313839154279369 n21;几何和,11011715916118384213241289333368 n12;吉布斯和,14014317779187212333377 n17; Glenlair 和,118123137167366 n10,367 n9;grad 和,152243264332;Grassmann 和,368 n12;重力和,12533142367 n11,368 n12,368 n15 ; Hamilton和,1181912813745368 n12;健康,166;海维赛德和,169 78 187 212 333 369 n21,372 nn2–10,375 n10,376 nn11–15 虚数和,143;无穷小和,122 积分和,1223313740367 n8,368 n12,368 n15,369 n19,369 n21 ;平方反比定律和,125138367 n11,368 n15;拉格朗日和,125 30368 n12 拉普拉斯,126 30144369 n21 运动定律和,125;莱布尼茨和,126,368 n12;光和,16,18,35,14142,173,19820 0,212,332,349 n1,387 n10 ;“信念写下字” 121 磁场矢量118,155,2012,264,287 运动 125,129,145 ;牛顿和,125,128,130,133-37,142,367 n11,368 n12,368 nn14-15 ;符号140,143,149-50,152-57 行星133,142,199诗歌和,137,165,231;肖像,231 3,7,​​11011 , 11722 , 128 , 14246 , 14987 , 243 , 325 , 33233 , 365 n14 , 372 nn4–10, 373 n11, 373 n15, 375 n10, 376 nn11–14, 377 n15 , 377 n17 ;无线电波和,xii 16,141,159,175,241,333 现实和,130;“关于物理量分类评论” ,151数和,13031,140,143 – 44 萨默维尔和,126时空和, 19520 7 212 15 380 nn11–12;雕像,158;斯托克定理和,12113839152157160367 n7,369 n21,370 n21,380 n11 ;泰特118 – 24 ,1282913713914245152167331366 n2,367 n7,369 n18,369 n21 和,238,24143,256,264,271,273,27980,287 90,300,387 n10 ;汤姆森 122 – 24,129,131,136,140,142,367 n7 时间线论,331 34横波14142论文145 另请参阅电磁学论文;​​矢量3,110,124,139-42,150-55,157,159,166,183,201-4,215,237,264,287-89,325,331-32,343n2 速度和,130-31,136 视觉 123 ;“牧马愿景” ,120-21​

Maxwell, James Clerk, xi; abstract concepts and, 118; action-at-a-distance vs. fields and, 13237; algebra and, 3, 7, 16; analogies and, 12930, 132, 13536, 15152; animals and, 120, 158, 160; Archimedes and, 122; astronomy and, 145; background of, 11819; calculus and, 18, 35, 349nn1–2; Campbell and, 119, 150, 153, 157, 16667; Cartesian coordinates and, 128, 143; Cayley and, 137; colour and, 123, 158, 248; Coloumb’s law and, 138, 154, 367n11, 370n21; convergence and, 153, 155, 161, 172, 372n7; curved space and, 219, 231, 23539; death of, 16667; De Morgan and, 145; derivatives and, 12627, 13738, 141, 152, 172, 264, 271, 300, 369n21, 376n14; differential calculus and, 127, 13741, 369n19, 369n21; divergence and, 144, 149, 15255, 161, 172, 243, 264, 332, 369n21, 372n7, 375n10; Einstein and, xii, 7, 35, 129, 141, 160, 183, 196, 200201, 205, 207, 212, 238, 24142, 280, 287, 289, 325, 334, 366n1, 380n12; electricity and, 12345, 150, 154, 157, 168, 332, 343n2, 349n2, 365n14, 367n11, 368n14, 369n19, 369n21, 372n7, 376n14; electromagnetism and, 3, 16, 110, 11846, 15051, 15660, 17073, 177, 183, 198204, 213, 241, 271, 280, 28890, 320, 325, 33233, 371n23, 376n11; electromotive intensity and, 153; electrons and, 12425, 129; Euler and, 136; Faraday and, 12829, 13342, 149, 151, 167, 237, 279, 289, 331, 369n21; flux and, 13033, 13738, 369nn20–21; Gauss and, 126, 130, 133, 13839, 154, 279, 369n21; geometry and, 110, 117, 159, 161, 18384, 213, 241, 289, 333, 368n12; Gibbs and, 140, 143, 17779, 187, 212, 333, 377n17; Glenlair and, 118, 123, 137, 167, 366n10, 367n9; grad and, 152, 243, 264, 332; Grassmann and, 368n12; gravity and, 12533, 142, 367n11, 368n12, 368n15; Hamilton and, 11819, 128, 13745, 368n12; health of, 166; Heaviside and, 16978, 187, 212, 333, 369n21, 372nn2–10, 375n10, 376nn11–15; imaginary numbers and, 143; infinitesimals and, 122; integrals and, 12233, 13740, 367n8, 368n12, 368n15, 369n19, 369n21; inverse square law and, 125, 138, 367n11, 368n15; Lagrange and, 12530, 368n12; Laplace and, 12630, 144, 369n21; laws of motion and, 125; Leibniz and, 126, 368n12; light and, 16, 18, 35, 14142, 173, 198200, 212, 332, 349n1, 387n10; “Lines Written under the Conviction,” 121; magnetic field vector and, 118, 155, 2012, 264, 287; motion and, 125, 129, 145; Newton and, 125, 128, 130, 13337, 142, 367n11, 368n12, 368nn14–15; notation and, 140, 143, 14950, 15257; planets and, 133, 142, 199; poetry and, 137, 165, 231; portrait of, 231; quaternions and, 3, 7, 11011, 11722, 128, 14246, 14987, 243, 325, 33233, 365n14, 372nn4–10, 373n11, 373n15, 375n10, 376nn11–14, 377n15, 377n17; radio waves and, xii, 16, 141, 159, 175, 241, 333; reality and, 130; “Remarks on the Classification of Physical Quantities,” 151; scalar numbers and, 13031, 140, 14344; Somerville and, 126; space-time and, 195207, 21215, 380nn11–12; statue of, 158; Stoke’s theorem and, 121, 13839, 152, 157, 160, 367n7, 369n21, 370n21, 380n11; Tait and, 11824, 12829, 137, 139, 14245, 152, 167, 331, 366n2, 367n7, 369n18, 369n21; tensors and, 238, 24143, 256, 264, 271, 273, 27980, 28790, 300, 387n10; Thomson and, 12224, 129, 131, 136, 140, 142, 367n7; timeline on, 33134; transverse waves and, 14142; Treatise, 145 (see also Treatise on Electricity and Magnetism); vector field and, 3, 110, 124, 13942, 15055, 157, 159, 166, 183, 2014, 215, 237, 264, 28789, 325, 33132, 343n2; velocity and, 13031, 136; vision and, 123; “A Vision of a Wrangler,” 12021

马克斯韦尔,凯瑟琳·杜瓦,137 , 14950

Maxwell, Katherine Dewar, 137, 14950

麦克斯韦学派, 16869

Maxwellians, 16869

麦克斯韦方程组:导数与积分,36971 n21;赫维赛德参见麦克斯韦的全矢量形式,15257 , 17217 6, 375 n10, 376 nn13–15;洛伦兹变换下的不变性,19920 4, 212;张量形式,215 , 264 , 28789另请参阅詹姆斯·克拉克·麦克斯韦

Maxwell’s equations: derivatives vs. integrals and, 36971n21; Heaviside cf. Maxwell whole vector form of, 15257, 172176, 375n10, 376nn13–15; invariance under Lorentz transformations of, 199204, 212; tensor form of, 215, 264, 28789. See also Maxwell, James Clerk

玛雅数学,xvii

Mayan mathematics, xvii

麦考利,亚历山大,18185

McAulay, Alexander, 18185

力学研究所,xix

Mechanics’ Institutes, xix

天体机制》(萨默维尔),65,330

Mechanism of the Heavens (Somerville), 65, 330

水星276,292-93,302,395n32​​

Mercury, 276, 29293, 302, 395n32

索不达米亚数学,xviixxvi;代数与,xiv9-11、13;粘土板,xiv323、327、343 n3、344 n4;几何与,xiv ;四数参见,186 ;空间与,99 ;张量参见,255;时间轴,327

Mesopotamian mathematics, xvii, xxvi; algebra and, xiv, 911, 13; clay tablets of, xiv, 323, 327, 343n3, 344n4; geometry and, xiv; quaternions cf., 186; space and, 99; tensors cf., 255; timeline on, 327

穷尽法,2223

method of exhaustion, 2223

迈克尔逊-莫雷实验,198 - 99 年207 年380 年n11

Michelson-Morley experiment, 19899, 207, 380n11

显微镜(以及广义相对论太空测试),275,397 n44

MICROSCOPE (and space-based tests of general relativity), 275, 397n44

《米德尔马契》(艾略特),16566

Middlemarch (Eliot), 16566

三重古斯塔夫, 296 , 318 , 395 n34

Mie, Gustav, 296, 318, 395n34

身心问题,164

mind-body problem, 164

夫斯基,赫尔曼xxii弯曲空间217,219,223,227,234,23839,383 n5,386 n16 ;逝世 289 爱因斯坦196,20712,217,219,238,242,28087,289,293,303,334,340,383 n5;第四维207 8希尔伯特 211,293 ;与,381 n20 时空与,19620712215219287289298334382 n27,392 n18,397 n39;张量与,242 – 43 ,253255268692808929329899303 392 n18,397 nn39–40 ;时间轴开启334

Minkowski, Hermann, xxii; curved space and, 217, 219, 223, 227, 234, 23839, 383n5, 386n16; death of, 289; Einstein and, 196, 20712, 217, 219, 238, 242, 28087, 289, 293, 303, 334, 340, 383n5; fourth dimension and, 2078; Hilbert and, 211, 293; imaginary numbers and, 381n20; space-time and, 196, 20712, 215, 219, 287, 289, 298, 334, 382n27, 392n18, 397n39; tensors and, 24243, 253, 255, 26869, 28089, 293, 29899, 303, 392n18, 397nn39–40; timeline on, 334

夫斯基度量:弯曲空间和,219,227,234,239;爱因斯坦和,196,207 12,217,219,238,242,280 – 87,289,293,303,334,340,383 n5 时空210 – 11 242 43,255,26869,280,28385,397 n40,400 n9​​​

Minkowski metric: curved space and, 219, 227, 234, 239; Einstein and, 196, 20712, 217, 219, 238, 242, 28087, 289, 293, 303, 334, 340, 383n5; space-time and, 21011; tensors and, 24243, 255, 26869, 280, 28385, 397n40, 400n9

莫比乌斯, 八月, 102 , 10912 , 227

Möbius, August, 102, 10912, 227

动量:角动量,949633536362 n30;守恒,16 ,291 ,299 – 30 0,305 爱因斯坦30511,321 矢量思想 43 45 353 n8;牛顿和,43452953068307373 n11;诺特定和,16;空间和,80819496;自旋和,9496335362 n30;张和,29091,295,298 – 30 0,321 ;时间轴开启334 36

momentum: angular, 9496, 33536, 362n30; conservation of, 16, 291, 299300, 305; Einstein and, 30511, 321; ideas for vectors and, 43, 45, 353n8; Newton and, 43, 45, 295, 3068, 307, 373n11; Noether and, 16; space and, 8081, 9496; spin and, 94, 96, 335, 362n30; tensors and, 29091, 295, 298300, 321; timeline on, 33436

单极子:重力磁单极子,155372 n9,385 n7,398 n44 ;单极子,155372 n9,376 n13

monopoles: gravitomagnetic, 155, 372n9, 385n7, 398n44; magnetic, 155, 372n9, 376n13

莫利,爱德华,19899 , 207 , 380 n11

Morley, Edward, 19899, 207, 380n11

运动:弹道12,47-49;布朗,347 n12;微积分27,30,34-36,39,41 碰撞50-52,328,353 n8;弯曲空间,217,219,230 ;爱因斯坦和,35,200-20 1,217,219,275,295-96,306-8,347 n12,379 n10,392 n12,395 n32,397 n44​下落112,125,129,219,275 – 76,277,304,308,352 n15,353 n6,390 n5,397 n40;重力,27,34 – 36,4649,125,182,219,27576,368 n15,397 n44 哈里奥特4849,112,353 n6,353 n8 ;矢量想法4352;拉格朗日和,105 麦克斯韦156,174 ​​​牛顿和,3436 , 39 , 4349 , 105 , 125 , 182 , 202 , 271 , 275 , 283 , 3068 , 325 , 351 n12, 358 n12, 368 n15, 392 n32;行星齿轮,3436 , 45 , 105 ,344 n6、351 n12 ;势能125159,172;空间和,73,93 ;时空和,189,199-20 2,205257-58,264,271-76,283,295-96,299;轨迹35,46-49,181-82,274,277 矢量 156,161,172,182

motion: ballistic, 12, 4749; Brownian, 347n12; calculus and, 27, 30, 3436, 39, 41; collisions, 5052, 328, 353n8; curved space and, 217, 219, 230; Einstein and, 35, 200201, 217, 219, 275, 29596, 3068, 347n12, 379n10, 392n12, 395n32, 397n44; falling, 112, 125, 129, 219, 27576, 277, 304, 308, 352n15, 353n6, 390n5, 397n40; gravity and, 27, 3436, 4649, 125, 182, 219, 27576, 368n15, 397n44; Harriot and, 4849, 112, 353n6, 353n8; ideas for vectors and, 4352; Lagrange and, 105; Maxwell and, 156, 174; Newton and, 3436, 39, 4349, 105, 125, 182, 202, 271, 275, 283, 3068, 325, 351n12, 358n12, 368n15, 392n32; planetary, 3436, 45, 105, 344n6, 351n12; potential energy and, 125; quaternions and, 159, 172; space and, 73, 93; space-time and, 189, 199202, 205; tensors and, 25758, 264, 27176, 283, 29596, 299; trajectories and, 35, 4649, 18182, 274, 277; vectors and, 156, 161, 172, 182

乘法表(美索不达米亚),xiv

multiplication tables (Mesopotamian), xiv

nabla:汉密尔顿和,1434414614815253243298332;符号和,14356161243264298332;泰特和,14344146148150156161243

nabla: Hamilton and, 14344, 146, 148, 15253, 243, 298, 332; notation and, 14356, 161, 243, 264, 298, 332; Tait and, 14344, 146, 148, 150, 156, 161, 243

拿破仑战争,220

Napoleonic Wars, 220

美国宇航局64、93、98、33637

NASA, 64, 93, 98, 33637

自然语言处理( NLP) ,24951,387 n12,388 n13

natural language processing (NLP), 24951, 387n12, 388n13

自然哲学:与现代理论物理学的比较,3435

natural philosophy: comparison with modern theoretical physics, 3435

自然杂志, 150 , 158 , 18081 , 185 , 281

Nature journal, 150, 158, 18081, 185, 281

纳粹,317

Nazis, 317

海王星,20

Neptune, 20

中子,94,97,362 n30,363 n31

neutrons, 94, 97, 362n30, 363n31

纽科门蒸汽机,157

Newcomen steam engine, 157

新思想时代,A(Hinton),206

New Era of Thought, A (Hinton), 206

新格兰奇,xvii

New Grange, xvii

牛顿,汉娜27-28

Newton, Hannah, 2728

牛顿艾萨克:代数7,14;算法323;背景,27 28 巴罗和,27 ;微积分和,1820,2641,323,350 nn6–7,351 nn11–13,352 n15数学家创造性作用和,3637,158弯曲空间和,23163,137,172,182,202,271,295,301,352 n15,368 n12,373 n11;爱因斯坦306 8,325 ;爱因斯坦基于牛顿方程式,27580 , 283 , 28587 , 295 , 397 n41;方程优雅325;作为天才26,42;几何和,14,39-41,44,60,62,74,308,329,350 n7,351 n15,368 n12 重力27,34-40,47,125,130,133,142,182,275-80,285-87,301-2,307,329,337,367 n11,368 n12,368 n15,391 nn7-8,397 n41;​​汉密尔顿和,42,52,60,62 – 65,74,84,103,146 – 47,181,196,306,325,355 n20,358 n12,367 n12 胡克 36 37,351 nn12–13,373 n15 矢量思想,42 54,57 – 65,111 无穷2630平方反比定律34,36,125,277,279,329,367 n11;​​​​运动定律和,3436 , 4344 , 4849 , 125 , 182 , 202 , 271 , 275 , 283 , 306 , 358 n12, 368 n15; 光和,1819 , 3637202277280285303337397 n44;麦克斯韦和,12512813013337142367 n11,368 n12,368 nn14–15;动量和,43452953068,373 n11 运动和,3436,39,4349,105,125,182,202,271,275,283,3068,325,351 n12,358 n12,368 n15,392 n32 符号,111 12,172,176数值线性代数(NLA ),323平行四边形规则,44 行星133;肖像,231 Principia3445,49,52,65,74,176,32829,350 n7,351 nn12–13,352 n15 四元数18182;宗教72;保密 27 – 28空间72,74,83 – 84,358 n12 时空xxixxii 196,201 – 2 295,395 n32,397 n41,397 n44 ;时间轴328 – 29,331,337​​​

Newton, Isaac: algebra and, 7, 14; algorithms and, 323; background of, 2728; Barrow and, 27; calculus and, 1820, 2641, 323, 350nn6–7, 351nn11–13, 352n15; creative role of mathematicians and, 3637, 158; curved space and, 231; derivatives and, 63, 137, 172, 182, 202, 271, 295, 301, 352n15, 368n12, 373n11; Einstein and, 3068, 325; Einstein’s equations based on Newton’s, 27580, 283, 28587, 295, 397n41; elegance of equation and, 325; as genius, 26, 42; geometry and, 14, 3941, 44, 60, 62, 74, 308, 329, 350n7, 351n15, 368n12; gravity and, 27, 3440, 47, 125, 130, 133, 142, 182, 27580, 28587, 3012, 307, 329, 337, 367n11, 368n12, 368n15, 391nn7–8, 397n41; Hamilton and, 42, 52, 60, 6265, 74, 84, 103, 14647, 181, 196, 306, 325, 355n20, 358n12, 367n12; Hooke and, 3637, 351nn12–13, 373n15; ideas for vectors and, 4254, 5765, 111; infinitesimals and, 2630; inverse square law and, 34, 36, 125, 277, 279, 329, 367n11; laws of motion and, 3436, 4344, 4849, 125, 182, 202, 271, 275, 283, 306, 358n12, 368n15; light and, 1819, 3637, 202, 277, 280, 285, 303, 337, 397n44; Maxwell and, 125, 128, 130, 13337, 142, 367n11, 368n12, 368nn14–15; momentum and, 43, 45, 295, 3068, 373n11; motion and, 3436, 39, 4349, 105, 125, 182, 202, 271, 275, 283, 3068, 325, 351n12, 358n12, 368n15, 392n32; notation and, 11112, 172, 176; numerical linear algebra (NLA), 323; parallelogram rule and, 44; planets and, 133; portrait of, 231; Principia, 3445, 49, 52, 65, 74, 176, 32829, 350n7, 351nn12–13, 352n15; quaternions and, 18182; religion and, 72; secrecy of, 2728; space and, 72, 74, 8384, 358n12; space-time and, xxixxii, 196, 2012; tensors and, 295, 395n32, 397n41, 397n44; timeline on, 32829, 331, 337

《纽约时报》(宣布验证了爱因斯坦的光弯曲预测),303

New York Times (announcing verification of Einstein’s light-bending prediction), 303

九章算术(中文版)8283

Nine Chapters on the Mathematical Art (Chinese text), 8283

诺贝尔奖,198,200,228,337,385 n7,402 n2

Nobel Prize, 198, 200, 228, 337, 385n7, 402n2

无克隆定理,321

no-cloning theorem, 321

Noether, Emmy: 代数和, 7 , 16 ; Bianchi 恒等式和, 31011 , 336 , 401 n13; 能量动量守恒定律和, 3069 ; 爱因斯坦和, 30413 , 317 , 399 n3, 399 n5; 广义相对论和, 3045 ; 希尔伯特和, 3046 , 31013 , 335 ; 不变性和, 3068 , 399 n5; “不变变分问题”, 306 ; 克莱因和, 3046 , 310 , 31213 , 335 , 381 n14;动量和,16,306-9作为现代代数母,7;狭义相对论和,308,312;争取接受,312-14;漩涡321 时间轴335-36

Noether, Emmy: algebra and, 7, 16; Bianchi identities and, 31011, 336, 401n13; conservation of energy-momentum and, 3069; Einstein and, 30413, 317, 399n3, 399n5; general theory of relativity and, 3045; Hilbert and, 3046, 31013, 335; invariance and, 3068, 399n5; “Invariante Variationsprobleme,” 306; Klein and, 3046, 310, 31213, 335, 381n14; momentum and, 16, 3069; as mother of modern algebra, 7; special theory of relativity and, 308, 312; struggle for acceptance by, 31214; Swirles and, 321; timeline on, 33536

Noether定理3067,311,33536,400n7

Noether theorems, 3067, 311, 33536, 400n7

欧几里得几何,117,163,216,220,284,330

non-Euclidean geometry, 117, 163, 216, 220, 284, 330

北大西洋电报公司,148

North Atlantic Telegraph Company, 148

北不列颠评论 (期刊——泰特为汉密尔顿写的讣告),146

North British Review (journal— Tait’s obituary for Hamilton), 146

诺顿,约翰 D.,399n2

Norton, John D.,399n2

符号:代数和,1112、15积分和,29、32、39 40 ;笛卡尔坐标和,xxiixxiii;克里斯托费尔和,237 39,271 73,300,303,386 n15,397 n40,397 n42 克利福德和,161 62 弯曲空间和,22223,233 – 39,383 n6 ;发展xxxxvi;狄拉克和,251爱因斯坦30030 1 ;欧拉和,5253;高斯和,383 n6;几何和,xxiixxiii德文字母,153,155 ;吉布斯78,376 n12 ;希腊字母,29,114,143,147,153,178,234,261,271;Hamilton373 n12 Harriot和,9,14 – 15,17,31,51,35354 n8 Heaviside78,143,17273 矢量思想和,52,62,353 n8 索引23536,253 – 54,258 – 71 Klein,222 Leibniz和,111 12,368 n12 麦克斯韦和,1401431495015257;nabla 和,14356161243264298332;牛顿和,11112;名称的力量和,15257;四元数和,1061111471525716117282185187;里奇和,298;黎曼和,23339383 n6;空间和,78;时空和,19620220721112215;张量和,23339251542586627328889298390 n23,400 ;时间轴上332334

notation: algebra and, 1112, 15; calculus and, 29, 32, 3940; Cartesian coordinates and, xxiixxiii; Christoffel and, 23739, 27173, 300, 303, 386n15, 397n40, 397n42; Clifford and, 16162; curved space and, 22223, 23339, 383n6; development of, xxxxvi; Dirac and, 251; Einstein and, 300301; Euler and, 5253; Gauss and, 383n6; geometry and, xxiixxiii; German letters, 153, 155; Gibbs and, 78, 376n12; Greek letters, 29, 114, 143, 147, 153, 178, 234, 261, 271; Hamilton and, 373n12; Harriot and, 9, 1415, 17, 31, 51, 35354n8; Heaviside and, 78, 143, 17273; ideas for vectors and, 52, 62, 353n8; index, 23536, 25354, 25871; Klein and, 222; Leibniz and, 11112, 368n12; Maxwell and, 140, 143, 14950, 15257; nabla and, 14356, 161, 243, 264, 298, 332; Newton and, 11112; power of names and, 15257; quaternions and, 106, 111, 147, 15257, 161, 17282, 185, 187; Ricci and, 298; Riemann and, 23339, 383n6; space and, 78; space-time and, 196, 202, 207, 21112, 215; tensors and, 23339, 25154, 25866, 273, 28889, 298, 390n23, 400; timeline on, 332, 334

数值线性代数(NLA),323

numerical linear algebra (NLA), 323

数,99,363n34

octonions, 99, 363n34

日全食下的月亮颂歌(汉密尔顿),98

Ode to the Moon under Total Eclipse (Hamilton), 98

Øersted ,Hans :电磁学和,94,110,124,128 - 29,132,138,173,330 法拉第 128 - 29,173,330 ;电报94 - 95,128,330

Øersted, Hans: electromagnetism and, 94, 110, 124, 12829, 132, 138, 173, 330; Faraday and, 12829, 173, 330; telegraphy and, 9495, 128, 330

“论法拉第的力线” (法拉第),136

“On Faraday’s Lines of Forces” (Faraday), 136

论运动物体的电动力学(爱因斯坦),334

On the Electrodynamics of Moving Bodies (Einstein), 334

论几何学基础的假设(克利福德),238

On the Hypotheses Which Lie at the Bases of Geometry (Clifford), 238

杰夫·奥帕特, 36 , 97 , 363 n31

Opat, Geoff, 36, 97, 363n31

OpenAI,249

OpenAI, 249

镊,20,290,349 n2​

optical tweezers, 20, 290, 349n2

光学,18,20,27,123,147

optics, 18, 20, 27, 123, 147

奥雷斯姆,妮可,xix27岁

Oresme, Nicole, xix, 27

奥斯特拉格拉德斯基,米哈伊尔,139

Ostragradsky, Mikhail, 139

奥斯曼人,47

Ottomans, 47

牛津英语词典,26,45,130

Oxford English Dictionary, 26, 45, 130

佩奇,拉里87,336

Page, Larry, 87, 336

PageRank ,8788,336,360 n21

PageRank, 8788, 336, 360n21

蓓尔迈尔公报167

Pall Mall Gazette, 167

抛物线, 21 , 49 , 82 , 182 , 277

parabolas, 21, 49, 82, 182, 277

平行四边形规则:弯曲空间和,220,315 ;爱因斯坦和,314 广义相对论和,314-19;格拉斯曼和,103-4;向量的思想和,44-52,65,352 n2,353 n8 牛顿和,44;张245,255时间轴328

parallelogram rule: curved space and, 220, 315; Einstein and, 314; general theory of relativity and, 31419; Grassmann and, 1034; ideas for vectors and, 4452, 65, 352n2, 353n8; Newton and, 44; tensors and, 245, 255; timeline on, 328

巴黎科学院,233

Parisian Academy of Sciences, 233

顾客,8、12、28、31、47-48、50

patrons, 8, 12, 28, 31, 4748, 50

沃尔夫冈·泡利, 96 , 336 , 362 nn29–30

Pauli, Wolfgang, 96, 336, 362nn29–30

乔治·皮科克 (George Peacock),356 n2;代数和,61627099100356 n2;汉密尔顿和,101;向量的思想和,616265;空间和,7099100356 n2

Peacock, George, 356n2; algebra and, 6162, 70, 99100, 356n2; Hamilton and, 101; ideas for vectors and, 6162, 65; space and, 70, 99100, 356n2

朱塞佩·皮亚诺,163

Peano, Giuseppe, 163

皮尔斯,本杰明,86岁

Peirce, Benjamin, 86

查尔斯·皮尔斯,86 , 178

Peirce, Charles, 86, 178

彭罗斯,罗杰228,237

Penrose, Roger, 228, 237

波斯数学,5,9,64,347 n15

Persian mathematics, 5, 9, 64, 347n15

彼得·潘(巴里),145

Peter Pan (Barrie), 145

哲学杂志181

Philosophical Magazine, 181

《哲学学报》(皇家学会),256

Philosophical Transactions (Royal Society), 256

光子,1920,99,277,284,348 n17,361 n26,362 n30​

photons, 1920, 99, 277, 284, 348n17, 361n26, 362n30

音调:多普勒效应和声音,278;旋转和,9293

pitch: Doppler effect and sound, 278; rotations and, 9293

普朗克马克斯,81,292-93,362 n27

Planck, Max, 81, 29293, 362n27

行星:微积分和,3436;引力和,343613327576367 n11,393 n21;矢量的想法和,4445;麦克斯韦和,133142199;运动,343645105344 n6,351 n12;牛顿和,34 36 133 近日点/张量的需要和,27576

planets: calculus and, 3436; gravity and, 3436, 133, 27576, 367n11, 393n21; ideas for vectors and, 4445; Maxwell and, 133, 142, 199; motion of, 3436, 45, 105, 344n6, 351n12; Newton and, 3436, 133; perihelion/need for tensors and, 27576

普林普顿泥板(毕达哥拉斯三元),xvxvi57、186、327、343 n3、344 n4

Plimpton tablet (Pythagorean triples), xv, xvi, 57, 186, 327, 343n3, 344n4

庞加莱,亨利:背景,200;弯曲空间与,218;爱因斯坦与,2001205921221824228081334;以太与,200 20 1,205280334;洛伦兹与,19920 1,205 – 9212242280 334 时空与,2119920 1,205 – 9;时间轴334

Poincaré, Henri: background of, 200; curved space and, 218; Einstein and, 2001, 2059, 212, 218, 242, 28081, 334; ether and, 200201, 205, 280, 334; Lorentz and, 199201, 2059, 212, 242, 280, 334; space-time and, 21, 199201, 2059; timeline on, 334

泊松,西蒙-丹尼斯,12627 , 130 , 369 n21, 391 n7

Poisson, Siméon-Dénis, 12627, 130, 369n21, 391n7

多聚体,106

polyplets, 106

正电子, 32021

positrons, 32021

势(数学概念及应用),12528,130,137,143 – 44,152,157,172 – 73,176,270,27980,282,285 86,3078,329,368 n12,370 n21,373 n11–12,376 n11,391 n7

potential (mathematical concept and application of ), 12528, 130, 137, 14344, 152, 157, 17273, 176, 270, 27980, 282, 28586, 3078, 329, 368n12, 370n21, 373n11–12, 376n11, 391n7

势能,36,125,157,306,368n12,399n4

potential energy, 36, 125, 157, 306, 368n12, 399n4

庞德,罗伯特,336

Pound, Robert, 336

普雷格,谢丽尔,7岁

Praeger, Cheryl, 7

数学原理(du Châtelet),41

Principes Mathématiques (du Châtelet), 41

Principia自然哲学的数学原理)(牛顿):微积分和,34 41,176,350 n7,351 nn12–13,352 n15 向量思想和,42 45,49,52,65 ;接收3541空间和,74 时间轴32829

Principia (Mathematical Principles of Natural Philosophy) (Newton): calculus and, 3441, 176, 350n7, 351nn12–13, 352n15; ideas for vectors and, 4245, 49, 52, 65; reception of, 3541; space and, 74; timeline on, 32829

等效原理27678,282 85

principle of equivalence, 27678, 28285

生命与心灵问题(刘易斯),164

Problems of Life and Mind (Lewes), 164

新教徒,48 , 50

Protestants, 48, 50

质子,94,155,362n30

protons, 94, 155, 362n30

普鲁士220,241,296

Prussia, 220, 241, 296

托勒密,克劳狄斯,344 n6;《天文学大成》xvixviii5,327 ; 天文学大成》,xviii《地理学》,xviii 327 《向量的思想和》,4546

Ptolemy, Claudius, 344n6; Almagest, xvixviii, 5, 327; astronomy and, xviii; Geography, xviii, 327; ideas for vectors and, 4546

普京,弗拉基米尔,317

Putin, Vladimir, 317

毕达哥拉斯定理,xv ,45,22,23,5455,60,208,231,243,327,344 n7,382 n24 代数 4 5,346 n5 ;背景xv;微积分22,23弯曲空间231向量的思想和,5455,60普林普顿石板和,xv 186,327,344 n4 时空 208,382 n24和,243 时间轴327

Pythagoras’s theorem, xv, 45, 22, 23, 5455, 60, 208, 231, 243, 327, 344n7, 382n24; algebra and, 45, 346n5; background of, xv; calculus and, 22, 23; curved space and, 231; ideas for vectors and, 5455, 60; Plimpton tablets and, xv, 186, 327, 344n4; space-time and, 208, 382n24; tensors and, 243; timeline on, 327

勾股数,xv186

Pythagorean triples, xv, 186

Quach, James(和量子温度计),389 n20

Quach, James (and quantum thermometer), 389n20

二次方程:代数和,67,10,13,1516,347 n16,348 n19 ;弯曲空间和,23338 ;向量想法53,354 n11 ;空间和82时空 194,208,378 n5,381 n20,243,257 ;时间轴327

quadratic equations: algebra and, 67, 10, 13, 1516, 347n16, 348n19; curved space and, 23338; ideas for vectors and, 53, 354n11; space and, 82; space-time and, 194, 208, 378n5, 381n20; tensors and, 243, 257; timeline on, 327

量子计算机,xiv251-52,321,388n14

quantum computers, xiv, 25152, 321, 388n14

量子电动力学(QED),32021

quantum electrodynamics (QED), 32021

量子场论,320

quantum field theory, 320

量子力学:爱因斯坦和,311,320-21,323 ;向量想法55,66 空间和75,80,92-98,363 n31量和,247-55 ;时间轴336

quantum mechanics: Einstein and, 311, 32021, 323; ideas for vectors and, 55, 66; space and, 75, 80, 9298, 363n31; tensors and, 24755; timeline on, 336

量子理论,xiv ;代数14,323,347 n12 微积分和,19;狄拉克96,155,251,32021,336,362 n29,376 n13,402 n2;爱因斯坦19,292,302,311,347 n12 ;规范176;矢量思想55,66;光和,19,199,347单极155,376 n13 不可克隆定理 321 ;光子和1920,99,277,284,348 n17,361 n26,362 n30 普朗克81,29293,362 n27 ;薛定谔 55,66,75,323,348 n17 ;空间75,80,9298,362 n30,363 n31 ;时空和,199 自旋,94 – 98,175,251,292,318,321,33536,362 nn27–30,363 n31 ​​​张量和,24755,265,292,302,388 n14,389 n20 ;时间轴开启336

quantum theory, xiv; algebra and, 14, 323, 347n12; calculus and, 19; Dirac and, 96, 155, 251, 32021, 336, 362n29, 376n13, 402n2; Einstein and, 19, 292, 302, 311, 347n12; gauges and, 176; ideas for vectors and, 55, 66; light and, 19, 199, 347; monopoles and, 155, 376n13; no-cloning theorem and, 321; photons and, 1920, 99, 277, 284, 348n17, 361n26, 362n30; Planck and, 81, 29293, 362n27; Schrödinger and, 55, 66, 75, 323, 348n17; space and, 75, 80, 9298, 362n30, 363n31; space-time and, 199; spin and, 9498, 175, 251, 292, 318, 321, 33536, 362nn27–30, 363n31; tensors and, 24755, 265, 292, 302, 388n14, 389n20; timeline on, 336

夸克,99

quarks, 99

抽象概念和107,110,114,116,163;代数和,3,7 8,14,17,76 – 80,1029,11216,151,16063,17785,345 n1,346 n4 算法和,108,186;算术104 天文学和,102,114违反规则和, 8081 布鲁姆桥涂鸦 1 2,97,145,243,325,331 Campbell 和, 150 , 153 , 157 , 16667 ;笛卡尔坐标和, 150 , 184 , 186 ; Cayley 和, 8 , 8183 , 86 , 90 , 179 , 187 , 194 , 206 , 33233 , 359 n15; Clifford 和, 16067 , 172 , 177 , 179 , 181 ;交换律和, 104 , 107 , 11416 , 16263 , 178 , 332 , 356 n2;复数和104,107,109,114,116,178;收敛和,31,153 – 55,161,172 度和,152,156 – 57,172 – 75,370n21,373 n11,376 n12,376 n14 ;弯曲空间和, 229 ;曲面和, 111,115 ;德·摩根, 104,106,112-14,151,160,163,183 ;, 152,172-75,182 ;笛卡尔和, 112,151 ;积分和 , 105,109,111,371 n23 ;狄拉克, 155,376 n13 ;发散, 152-57,161,172-74 ;​爱因斯坦和,116,160,167,180,183 电和150,154,157,168-69,185,195 电磁学和 110,146,150-51,156-60,168,170-73,176-77,183,185,371 n23,372 n9,373 n11,376 n11,376 n13 电子 155,165,175 欧几里得和,11517、163、179;欧拉105;法拉第和,147、149、151、167、173 74 ​​通量151,154,155,174,375 n10;四维几何 187 ;伽利略112高斯和,108 – 11,115,154,174 ;广义相对论 160,176,372 n9,373 n12 几何 103 6,10917,15963,181 – 85,374 n23 吉布斯17781,18487,376 n12,377 nn17–18 ;万向和,93;​ grad and, 152 , 376 n14; 逐步接受, 16063 ; Grassmann and, 10218 , 126 , 16163 , 17881 , 364 n3, 364 n8, 365 n14, 366 n15, 366 n19, 374 n23, 377 n17; 重力和, 155 , 182 ;汉密尔顿和,3,7 - 8,14,17,64 - 65,71,74 - 83,86,90 - 91,97 - 107,101 - 19,128,142 - 53,159 - 63,171,178 - 86,196,206,229,243 - 44,325,331 - 33,336,355 n20,359 n14,360 n24,364n3,366 nn18–19,372 n10Harriot11112; Heaviside和153,16981,18487,375 nn2–10,376 nn11–13,377 n15 ;向量想法64 – 65,355 n20 ;2,61,75,78,153,171,187积分15152,373 n11;逆162,178,360 n24,362 n30,374 n22 ;​平方反比定律和,154;拉格朗日和,1045147364 nn6–7;拉普拉斯和, 1045四元数讲座76107114117119142260 n24,345 n1,355 n20,357 n4 ;莱布尼茨和,10511112366 n15;洛伦兹和,175;矩阵和,162362 n29 麦克斯韦和,3、7、110-11、117-22、128、142-87、243、325、332-33、365 n14、372 nn4-10、373 n11、373 n15、375 n10、376 nn11-14、377 n15、377 n17 不达米亚数据存储参见,186 ;作为思维方法158-59 运动和,103、105、112、117、156、159​​​, 161 , 164 , 172 , 182 ; nabla 和, 14356 , 161 , 243 , 264 , 298 , 332 ; 牛顿和, 1035 , 111 , 14647 , 158 , 172 , 17677 , 18182 , 365 n14, 373 n11, 373 n15; 符号和, 106 , 111 , 147 , 15257 , 161 , 17282 , 185 , 187 ; 平行四边形规则和, 1034 ;利矩阵336,362 n29 名称 幂15257数学原理和,176;量子理论和97,155,176,376 n13 无线电波159,175 现实176114,116,163,178机器人和,187;旋转,104 – 5,107,114,156,159,162,174,178,183,360 n24,362 n30 旋转3,90-94,97,104-7,114,159,162,178,183,267,336,345 n1,360 n24,362 n30量和,79,106-7,113,147-48151571616317117782185;萨默维尔和,765164;空间和,71748386909497100359 n14 ;时空和,189192194962067210;狭义相对论和,160;自旋和,9498;对称性和,17476375 n11,376 n13; Tait 和,114117146671701778118487;张量和,24344258267;Thomson 和,14751155158591687217781186;三维空间和104,107,114,161-62,178,183 时间轴,328,331-33,336-37 矢量场和 152-55,159,175 ;矢量部分,79 速度146,156,172,182 沃利斯和,111 ;战争结束,179-86

quaternions: abstract concepts and, 107, 110, 114, 116, 163; algebra and, 3, 78, 14, 17, 7680, 1029, 11216, 151, 16063, 17785, 345n1, 346n4; algorithms and, 108, 186; arithmetic and, 104; astronomy and, 102, 114; breaking rules and, 8081; Broome Bridge graffiti and, 12, 97, 145, 243, 325, 331; Campbell and, 150, 153, 157, 16667; Cartesian coordinates and, 150, 184, 186; Cayley and, 8, 8183, 86, 90, 179, 187, 194, 206, 33233, 359n15; Clifford and, 16067, 172, 177, 179, 181; commutative law and, 104, 107, 11416, 16263, 178, 332, 356n2; complex numbers and, 104, 107, 109, 114, 116, 178; convergence and, 31, 15355, 161, 172; curl and, 152, 15657, 17275, 370n21, 373n11, 376n12, 376n14; curved space and, 229; curved surfaces and, 111, 115; De Morgan and, 104, 106, 11214, 151, 160, 163, 183; derivatives and, 152, 17275, 182; Descartes and, 112, 151; differential calculus and, 105, 109, 111, 371n23; Dirac and, 155, 376n13; divergence and, 15257, 161, 17274; Einstein and, 116, 160, 167, 180, 183; electricity and, 150, 154, 157, 16869, 185, 195; electromagnetism and, 110, 146, 15051, 15660, 168, 17073, 17677, 183, 185, 371n23, 372n9, 373n11, 376n11, 376n13; electrons and, 155, 165, 175; Euclid and, 11517, 163, 179; Euler and, 105; Faraday and, 147, 149, 151, 167, 17374; flux and, 151, 154, 155, 174, 375n10; four-dimensional geometry and, 187; Galileo and, 112; Gauss and, 10811, 115, 154, 174; general theory of relativity and, 160, 176, 372n9, 373n12; geometry and, 1036, 10917, 15963, 18185, 374n23; Gibbs and, 17781, 18487, 376n12, 377nn17–18; gimbal lock and, 93; grad and, 152, 376n14; gradual acceptance of, 16063; Grassmann and, 10218, 126, 16163, 17881, 364n3, 364n8, 365n14, 366n15, 366n19, 374n23, 377n17; gravity and, 155, 182; Hamilton and, 3, 78, 14, 17, 6465, 71, 7483, 86, 9091, 97107, 10119, 128, 14253, 15963, 171, 17886, 196, 206, 229, 24344, 325, 33133, 336, 355n20, 359n14, 360n24, 364n3, 366nn18–19, 372n10; Harriot and, 11112; Heaviside and, 153, 16981, 18487, 375nn2–10, 376nn11–13, 377n15; ideas for vectors and, 6465, 355n20; imaginary numbers and, 2, 61, 75, 78, 153, 171, 187; integrals and, 15152, 373n11; inverse of, 162, 178, 360n24, 362n30, 374n22; inverse square law and, 154; Lagrange and, 1045, 147, 364nn6–7; Laplace and, 1045; Lectures on Quaternions, 76, 107, 114, 117, 119, 142, 260n24, 345n1, 355n20, 357n4; Leibniz and, 105, 11112, 366n15; Lorentz and, 175; matrices and, 162, 362n29; Maxwell and, 3, 7, 11011, 11722, 128, 14287, 243, 325, 33233, 365n14, 372nn4–10, 373n11, 373n15, 375n10, 376nn11–14, 377n15, 377n17; Mesopotamian data storage cf., 186; as method of thinking, 15859; motion and, 103, 105, 112, 117, 156, 159, 161, 164, 172, 182; nabla and, 14356, 161, 243, 264, 298, 332; Newton and, 1035, 111, 14647, 158, 172, 17677, 18182, 365n14, 373n11, 373n15; notation and, 106, 111, 147, 15257, 161, 17282, 185, 187; parallelogram rule and, 1034; Pauli matrices and, 336, 362n29; power of names and, 15257; Principia and, 176; quantum theory and, 97, 155, 176, 376n13; radio waves and, 159, 175; reality and, 176; real numbers and, 114, 116, 163, 178; robots and, 187; rotation and, 1045, 107, 114, 156, 159, 162, 174, 178, 183, 360n24, 362n30; rotations and, 3, 9094, 97, 1047, 114, 159, 162, 178, 183, 267, 336, 345n1, 360n24, 362n30; scalar numbers and, 79, 1067, 113, 14748, 15157, 16163, 171, 17782, 185; Somerville and, 7, 65, 164; space and, 71, 7483, 86, 9094, 97100, 359n14; space-time and, 189, 192, 19496, 2067, 210; special theory of relativity and, 160; spin and, 9498; symmetry and, 17476, 375n11, 376n13; Tait and, 114, 117, 14667, 170, 17781, 18487; tensors and, 24344, 258, 267; Thomson and, 14751, 155, 15859, 16872, 17781, 186; three-dimensional space and, 104, 107, 114, 16162, 178, 183; timeline on, 328, 33133, 33637; vector field and, 15255, 159, 175; vector part of, 79; velocity and, 146, 156, 172, 182; Wallis and, 111; “wars” over, 17986

“四元数和向量代数”(Gibbs),18586

“Quaternions and the Algebra of Vectors” (Gibbs), 18586

量子比特,xiv251-52,321,388 n14​

qubits, xiv, 25152, 321, 388n14

皇后学院33,122

Queen’s College, 33, 122

Questiones Mechanicae(亚里士多德学派), 4547

Questiones Mechanicae (Aristotle’s school), 4547

无线电波:代数和,16 积分和,20赫兹159,175 麦克斯韦和,xii 16,141,159,175,241,333 空间和,98 量和,241 ;时间轴 333,337

radio waves: algebra and, 16; calculus and, 20; Hertz and, 159, 175; Maxwell and, xii, 16, 141, 159, 175, 241, 333; space and, 98; tensors and, 241; timeline on, 333, 337

罗利(Ralegh), 沃尔特, 8 , 31

Raleigh (Ralegh), Walter, 8, 31

拉姆勒,露丝,312

Ramler, Ruth, 312

现实:代数和,4,17;理解的突破,ix;麦克斯韦和,130 ;四元/

reality: algebra and, 4, 17; breakthroughs in understanding, ix; Maxwell and, 130; quaternions/

向量和,176;空间和,99100;时空和,210;张量和,271

vectors and, 176; space and, 99100; space-time and, 210; tensors and, 271

实数,xxi 代数23;向量概念,58、59、61、63、354 n11元数114、116、163、178空间和,69、71、74、76 77、361 n24

real numbers, xxi; algebra and, 23; ideas for vectors and, 58, 59, 61, 63, 354n11; quaternions and, 114, 116, 163, 178; space and, 69, 71, 74, 7677, 361n24

雷布卡,格伦,336

Rebka, Glen, 336

红移27881,336 37​

redshift, 27881, 33637

折射,18,20,66,102,331

refraction, 18, 20, 66, 102, 331

恐怖统治,105

Reign of Terror, 105

“关于物理量分类的评论”(麦克斯韦),151

“Remarks on the Classification of Physical Quantities” (Maxwell), 151

于尔根·雷恩, 395 n33, 396 n37

Renn, Jürgen, 395n33, 396n37

莱因德纸莎草书,349 n3

Rhind papyrus, 349n3

里奇,比安卡,243

Ricci, Bianca, 243

Ricci, Gregorio:背景,24042;弯曲空间和,2192022623739;爱因斯坦和,21930531031431619;Grossmann 和,21923824425325827528587290295303319334;Hamilton 和,244267298325;索引符号和,25864;不变性24244,25659,26263,26973,285,290,298 克莱242,256,274,334 列维-奇维塔和,219 20,274,285,303,316,319,334 ;符号和,298论文24243;皇家数学273;张积分和,241,244,27174,303,305,316,318,333 34 张量和24046,25375,28587,290,29495,298 – 30 3,320,323,325,333,393 n21,397 n42,400 n10 时间轴开启,333 – 34 ​​337 ;维罗纳和257-58

Ricci, Gregorio: background of, 24042; curved space and, 21920, 226, 23739; Einstein and, 219, 305, 310, 314, 31619; Grossmann and, 219, 238, 244, 253, 258, 275, 28587, 290, 295, 303, 319, 334; Hamilton and, 244, 267, 298, 325; index notation and, 25864; invariance and, 24244, 25659, 26263, 26973, 285, 290, 298; Klein and, 242, 256, 274, 334; Levi-Civita and, 21920, 274, 285, 303, 316, 319, 334; notation and, 298; papers of, 24243; Royal Mathematics Prize and, 273; tensor calculus and, 241, 244, 27174, 303, 305, 316, 318, 33334; tensors and, 24046, 25375, 28587, 290, 29495, 298303, 320, 323, 325, 333, 393n21, 397n42, 400n10; timeline on, 33334, 337; Veronese and, 25758

里鲍,乔治,13334

Riebau, George, 13334

Riemann , Bernhard 背景,230 Clifford 230 弯曲空间和,383 n6,385 nn10–11 曲面219,23036,244,269,383 n6,385 nn10–12,386 nn13–16 爱因斯坦31011 高斯和,219,230 – 33,235,243,269,303,332,383 n6 符号233 39,383 n6;张量和23339,243 – 44,248,255,257,260 – 61,267,269,273,283,28687,295,303,310 11,332,386 n9 时间轴开启,332

Riemann, Bernhard: background of, 230; Clifford and, 230; curved space and, 383n6, 385nn10–11; curved surfaces and, 219, 23036, 244, 269, 383n6, 385nn10–12, 386nn13–16; Einstein and, 31011; Gauss and, 219, 23033, 235, 243, 269, 303, 332, 383n6; notation and, 23339, 383n6; tensors and, 23339, 24344, 248, 255, 257, 26061, 267, 269, 273, 283, 28687, 295, 303, 31011, 332, 386n9; timeline on, 332

右手 2,79,155,156

right-hand rule, 2, 79, 155, 156

机器人:代数和,1,3 ;向量的想法,44;矩阵和,8690;四元数和,187;空间和,8690,92;时空和,190 张量和,259;时间轴336

robots: algebra and, 1, 3; ideas for vectors and, 44; matrices and, 8690; quaternions and, 187; space and, 8690, 92; space-time and, 190; tensors and, 259; timeline on, 336

罗布森,埃莉诺,347 n13

Robson, Eleanor, 347n13

滚动, 9293

roll, 9293

罗马数字,245

Roman numerals, 245

浪漫主义,1、38、64、73、84

Romanticism, 1, 38, 64, 73, 84

《自己的房间》(伍尔夫),189

“Room of One’s Own” (Woolf ), 189

旋转代数13,14;复数5860弯曲空间和,227;汉密尔顿1,3,14,54,60,6768,89 – 91,97,100,104,107,114,178,183,336,360 n24 向量想法5455,5960,6768俯仰9293 数和,3,9094,97,104 – 7,114,156,159,162,174,178,183,267,336,345 n1,360 n24,362 n30 滚动,92 – 93 空间和,70,75,80,8997,100,360 n24,362 n30,363 n31 ;时空192,203,209,214;自旋362量和,25969,389 nn21–22 三维,68,75,90 – 94,97,104,107,114,162,183,345 n1 时间轴开启 330,336 偏航 93

rotation: algebra and, 13, 14; complex numbers and, 5860; curved space and, 227; Hamilton and, 1, 3, 14, 54, 60, 6768, 8991, 97, 100, 104, 107, 114, 178, 183, 336, 360n24; ideas for vectors and, 5455, 5960, 6768; pitch, 9293; quaternions and, 3, 9094, 97, 1047, 114, 156, 159, 162, 174, 178, 183, 267, 336, 345n1, 360n24, 362n30; roll, 9293; space and, 70, 75, 80, 8997, 100, 360n24, 362n30, 363n31; space-time and, 192, 203, 209, 214; spin, 362; tensors and, 25969, 389nn21–22; three-dimensional, 68, 75, 9094, 97, 104, 107, 114, 162, 183, 345n1; timeline on, 330, 336; yaw, 93

Routh,EJ,121

Routh, E. J., 121

苏格兰皇家银行,7

Royal Bank of Scotland, 7

皇家运河,1

Royal Canal, 1

爱尔兰皇家学院82、98、117

Royal Irish Academy, 82, 98, 117

保皇党,27

Royalists, 27

皇家数学奖273,310

Royal Mathematics Prize, 273, 310

爱丁堡皇家学会98,159-60,194

Royal Society of Edinburgh, 98, 15960, 194

伦敦皇家学会139,160,256

Royal Society of London, 139, 160, 256

玛丽·萨德莱尔,188

Sadleir, Mary, 188

萨尔顿,格里,8687

Salton, Gerry, 8687

SARS-CoV-2 病毒, 191

SARS-CoV-2 virus, 191

标量:代数和,3积分 41 弯曲空间 225 26,234,236,383 n6,386 n13 点积,78,80,211 爱因斯坦401 n12 ;麦克斯韦13031,140,14344;符号 xx ;四元1067,113,147 – 48,151 – 57,16163,171,17782,185;空间和,7680 , 87 , 359 n14, 359 n20, 360 n24; 时空和,18993 , 21011 ; 张量和,243 , 246 , 25255 , 259 , 26270 , 27980 , 286 , 301 , 324 , 389 n20, 389 n22, 397 n39, 397 n43

scalar numbers: algebra and, 3; calculus and, 41; curved space and, 22526, 234, 236, 383n6, 386n13; dot products and, 78, 80, 211; Einstein and, 401n12; Maxwell and, 13031, 140, 14344; notation and, xx; quaternions and, 1067, 113, 14748, 15157, 16163, 171, 17782, 185; space and, 7680, 87, 359n14, 359n20, 360n24; space-time and, 18993, 21011; tensors and, 243, 246, 25255, 259, 26270, 27980, 286, 301, 324, 389n20, 389n22, 397n39, 397n43

斯考,一月,311、316、336

Schouten, Jan, 311, 316, 336

薛定谔,埃尔文55,66,75,323,348 n17​

Schrödinger, Erwin, 55, 66, 75, 323, 348n17

科学浪漫史(Hinton),206

Scientific Romances (Hinton), 206

科学家(新造术语),101

scientist (term coined), 101

斯科特,苏珊,313

Scott, Susan, 313

斯科特·沃尔特66 岁

Scott, Walter, 66

搜索引擎,8690,191,247,336 37,359n20​​​

search engines, 8690, 191, 247, 33637, 359n20

第二名:克利福德,160;麦克斯韦尔,121

Second Wrangler: Clifford, 160; Maxwell, 121

穆里尔·塞尔特曼,1617,349 n22

Seltman, Muriel, 1617, 349n22

资深牧马人:劳斯,121;泰特,119

Senior Wrangler: Routh, 121; Tait, 119

《理智与情感》(奥斯汀),145

Sense and Sensibility (Austen), 145

性别歧视,164,197,312,334

sexism, 164, 197, 312, 334

莎士比亚,威廉,49,149

Shakespeare, William, 49, 149

萧伯纳,乔治·伯纳德,167

Shaw, George Bernard, 167

信号处理,247

signal processing, 247

奇异值分解 (SVD), 86

singular value decomposition (SVD), 86

天狼星,十七

Sirius, xvii

热力学简图(Tait ),149,372 n3

Sketch of Thermodynamics (Tait), 149, 372n3

斜率,40,152,383n6

slope, 40, 152, 383n6

史密斯,巴纳巴斯,2728

Smith, Barnabas, 2728

史密斯奖,11922 , 138 , 183 , 256 , 367 n7

Smith’s Prize, 11922, 138, 183, 256, 367n7

社交媒体,360 n22

social media, 360n22

萨默维尔,玛丽:微积分和,19,38,349 n1;向量的思想和,6566;麦克斯韦和,126天体的机制65,330 ;四和,7,65,164;作为科学女王,7 空间和,73 时间轴,32930;杨和19

Somerville, Mary: calculus and, 19, 38, 349n1; ideas for vectors and, 6566; Maxwell and, 126; Mechanism of the Heavens, 65, 330; quaternions and, 7, 65, 164; as Queen of Science, 7; space and, 73; timeline on, 32930; Young and, 19

索末菲,阿诺德:弯曲空间和,219;爱因斯坦和,317;四矢量项,212;闵可夫斯基和,21112215219287289298334382 n27;时空和,21112215382 n27;张量和,212132879229529899;时间轴和,334;矢量委员会和,21112

Sommerfeld, Arnold: curved space and, 219; Einstein and, 317; four-vector term of, 212; Minkowski and, 21112, 215, 219, 287, 289, 298, 334, 382n27; space-time and, 21112, 215, 382n27; tensors and, 21213, 28792, 295, 29899; timeline and, 334; Vector Commission and, 21112

声波198,278,349 n1​

sound waves, 198, 278, 349n1

空间:抽象概念和,72,80;代数和,70 – 94,99 – 100;算法,82 – 83,87 89 算术71 72,76,356 n2人工智能AI 88 – 89 天文学73,76;违反规则和,80 81 笛卡尔坐标,77,82 凯莱和,8190交换律和76,7981,85,90,99;复数和,71,74 75,78,91,98 100,360 n24 弯曲,217 参见弯曲空间);德·摩根和,70 77,85,99 100,358 n9 狄拉克362 n29 爱因斯坦95电磁8384,98 – 99 ;电子93 – 100 ;欧几里得74欧拉和,72,81,90,356 n3,360 n24 四维数学和75 – 78,91,98 – 99 高斯和,8283,359 n16 广义相对论和,80 83,96 几何和74 – 75,87,90,360 n24 格拉斯曼和,196,212,215 引力92;汉密尔顿和,1 4,6886 , 8991 , 97100 , 1067 , 114 , 16162 , 183 , 356 n3, 357 n7, 358 n11, 359 nn13–14, 360 n24; 图像压缩和, 8690 ; 虚数和, 6970 , 7478 ; 运动定律和, 358 n12; 莱布尼茨和, 72 , 83 ; 洛伦兹和, 19921 2, 380 n12, 381 n14, 381 n20, 382 n24;矩阵和,71,81 93,96,360 n24,361 n25,362 nn29–30 达米亚和,99 ;动量和,8081,9496;运动和,73,93 ;牛顿和,72,74,8384,358 n12;符号,78 行星和,105数学原理74;二次方程82 量子理论和,75,80,92 – 98,362 n30,363 n31 数和,71,74-83,86,90-94,97-100,359 n14 无线电波和 98 现实,99-100 ;69,71,74,76-77,361 n24 机器人 86-90,92 旋转 70,75,80,89-​​97、100、360 n24、362 n30、363 n31;标量和7680、87、359 n14、359 n20、360 n24 ​​​搜索引擎和,86 90 萨默维尔和,73;对称性和,90;泰特和,18995,206;三维,74 另见三维空间)

space: abstract concepts and, 72, 80; algebra and, 7094, 99100; algorithms and, 8283, 8789; arithmetic and, 7172, 76, 356n2; artificial intelligence (AI) and, 8889; astronomy and, 73, 76; breaking rules and, 8081; Cartesian coordinates and, 77, 82; Cayley and, 8190; commutative law and, 76, 7981, 85, 90, 99; complex numbers and, 71, 7475, 78, 91, 98100, 360n24; curved, 217 (see also curved space); De Morgan and, 7077, 85, 99100, 358n9; Dirac and, 362n29; Einstein and, 95; electromagnetism and, 8384, 9899; electrons and, 93100; Euclid and, 74; Euler and, 72, 81, 90, 356n3, 360n24; four-dimensional mathematics and, 7578, 91, 9899; Gauss and, 8283, 359n16; general theory of relativity and, 8083, 96; geometry and, 7475, 87, 90, 360n24; Grassmann and, 196, 212, 215; gravity and, 92; Hamilton and, 14, 6886, 8991, 97100, 1067, 114, 16162, 183, 356n3, 357n7, 358n11, 359nn13–14, 360n24; image compression and, 8690; imaginary numbers and, 6970, 7478; laws of motion and, 358n12; Leibniz and, 72, 83; Lorentz and, 199212, 380n12, 381n14, 381n20, 382n24; matrices and, 71, 8193, 96, 360n24, 361n25, 362nn29–30; Mesopotamia and, 99; momentum and, 8081, 9496; motion and, 73, 93; Newton and, 72, 74, 8384, 358n12; notation and, 78; planets and, 105; Principia and, 74; quadratic equations and, 82; quantum theory and, 75, 80, 9298, 362n30, 363n31; quaternions and, 71, 7483, 86, 9094, 97100, 359n14; radio waves and, 98; reality and, 99100; real numbers and, 69, 71, 74, 7677, 361n24; robots and, 8690, 92; rotation and, 70, 75, 80, 8997, 100, 360n24, 362n30, 363n31; scalar numbers and, 7680, 87, 359n14, 359n20, 360n24; search engines and, 8690; Somerville and, 73; symmetry and, 90; Tait and, 18995, 206; three-dimensional, 74 (see also three-dimensional space)

时空:抽象概念和,206;代数和,194196207211;算法和,19192;人工智能(AI)和,191;天文学和,200;笛卡尔坐标和,190;凯莱和,18894206208;坐标变换和,1909419920 4,209 ;旋度和,204 211214;导数和,202345 n8;狄拉克和,96;发散和,211397 n39;爱因斯坦和xxiv19621 2,216,321 – 22,379 n10,381 n14 ;电磁学和,19820 4,213 电子200欧几里得209 – 10,216 四维,17,75,187,20512,248,289,298,381 n16,381 n18 广义相对论和196 97,206,210 12;​​​​几何和,190194 - 95206 - 7210 - 13216381 n14; 吉布斯和,196210212 ; 梯度和,211 ; 重力和,198 ; 希腊人和,90 ; 汉密尔顿和,381 n17; 海维赛德和,196206210 - 12 ; 不变性和,189 - 97202 - 5208212216378 n4,381 n14;克莱因和,212 莱布尼茨和,196;麦克斯韦和,195 20 7,212 15,380 nn11–12 可夫斯基和,19620712,215,334,382 n27,392 n18,397 n39运动189199 – 20 2,205 牛顿196 2012;符号和,xxi – xxii ,196202207211 – 12,215 庞加莱和,21,199 20 1,205 – 9 毕达哥拉斯和,208,382 n24;二次方程和,194,208,378 n5,381 n20 量子理论和,199;四元数和 189,192,194 96,206 7,210 现实和210;机器人190 旋转和,192,203,209,214;标189 93,21011 索末 211 12,215,382 n27 狭义相对论和,198 20 9 对称性和,19293,198,204,382 n27;汤姆森194,213 – 15 ;三维空间191 – 92,206 – 12,207,215 矢量 201 – 2;速度和,379 n10

space-time: abstract concepts and, 206; algebra and, 194, 196, 207, 211; algorithms and, 19192; artificial intelligence (AI) and, 191; astronomy and, 200; Cartesian coordinates and, 190; Cayley and, 18894, 206, 208; coordinate transformations and, 19094, 199204, 209; curl and, 204, 211, 214; derivatives and, 202, 345n8; Dirac and, 96; divergence and, 211, 397n39; Einstein and, xxiv, 196212, 216, 32122, 379n10, 381n14; electromagnetism and, 198204, 213; electrons and, 200; Euclid and, 20910, 216; four-dimensional, 17, 75, 187, 20512, 248, 289, 298, 381n16, 381n18; general theory of relativity and, 19697, 206, 21012; geometry and, 190, 19495, 2067, 21013, 216, 381n14; Gibbs and, 196, 210, 212; grad and, 211; gravity and, 198; Greeks and, 90; Hamilton and, 381n17; Heaviside and, 196, 206, 21012; invariance and, 18997, 2025, 208, 212, 216, 378n4, 381n14; Klein and, 212; Leibniz and, 196; Maxwell and, 195207, 21215, 380nn11–12; Minkowski and, 196, 20712, 215, 334, 382n27, 392n18, 397n39; motion and, 189, 199202, 205; Newton and, 196, 2012; notation and, xxixxii, 196, 202, 207, 21112, 215; Poincaré and, 21, 199201, 2059; Pythagoras and, 208, 382n24; quadratic equations and, 194, 208, 378n5, 381n20; quantum theory and, 199; quaternions and, 189, 192, 19496, 2067, 210; reality and, 210; robots and, 190; rotation and, 192, 203, 209, 214; scalar numbers and, 18993, 21011; Sommerfeld and, 21112, 215, 382n27; special theory of relativity and, 198209; symmetry and, 19293, 198, 204, 382n27; Thomson and, 194, 21315; three-dimensional space and, 19192, 20612, 207, 215; vector field and, 2012; velocity and, 379n10

狭义相对论,347 n12 弯曲空间与,232狄拉克320;以太与,198 20 1,205;引力与,198,219,275,278 – 81,285,291,397 n40,399 n5 诺特定与,308,312 与,160时空与,198 20 9 275,27885,29192,299

special theory of relativity, 347n12; curved space and, 232; Dirac and, 320; ether and, 198201, 205; gravity and, 198, 219, 275, 27881, 285, 291, 397n40, 399n5; Noether and, 308, 312; quaternions and, 160; space-time and, 198209; tensors and, 275, 27885, 29192, 299

光谱,96,336,361 n26

spectra, 96, 336, 361n26

自旋动量和,94,96,335,362 n30Gerlach和,95 – 96,335,362 n27 ;Goudsmit和,96,335,362 nn27–28;Lorentz96,362 n28 ;量子理论9498,175,251,292,318,321,33536,362 nn27–30,363 n31 94 98 Stern9596,292,335,362 n27 Uhlenbeck96,335,362 n27​

spin: angular momentum and, 94, 96, 335, 362n30; Gerlach and, 9596, 335, 362n27; Goudsmit and, 96, 335, 362nn27–28; Lorentz and, 96, 362n28; quantum theory and, 9498, 175, 251, 292, 318, 321, 33536, 362nn27–30, 363n31; quaternions and, 9498; Stern and, 9596, 292, 335, 362n27; Uhlenbeck and, 96, 335, 362n27

灵性121,137,164,206​​

spirituality, 121, 137, 164, 206

:空间和,76,280,294,303,321 – 22,337,361 n26,391 n8,393 n19,393 n21,397 n44 280,294,303,391 n8,393 n19,393 n21,397 n44​​​

stars: space and, 76, 280, 294, 303, 32122, 337, 361n26, 391n8, 393n19, 393n21, 397n44; tensors and, 280, 294, 303, 391n8, 393n19, 393n21, 397n44

静电,124128 - 30136138290329369 n21

static electricity, 124, 12830, 136, 138, 290, 329, 369n21

斯特恩,奥托,9596292335362 n27

Stern, Otto, 9596, 292, 335, 362n27

斯托克斯,乔治,122;矢量的分量形式,139;麦克斯韦理论和,160另请参阅斯托克斯定理

Stokes, George, 122; component form of vectors, 139; Maxwell’s theory and, 160. See also Stokes’s theorem

斯托克斯定理121,138 – 39,144,152,157,256,367 n7,369 n21,373 n11,388 n15 麦克斯韦和,121,138 39,152,157,160,367 n7,369 n21,370 n21,380 n11 史密斯 121 22,138

Stokes’s theorem, 121, 13839, 144, 152, 157, 256, 367n7, 369n21, 373n11, 388n15; Maxwell and, 121, 13839, 152, 157, 160, 367n7, 369n21, 370n21, 380n11; Smith’s Prize and, 12122, 138

巨石阵,xvii

Stonehenge, xvii

斯托特,艾丽西娅·布尔,206

Stott, Alicia Boole, 206

德克·斯特鲁克, 31112 , 316 , 336 , 401 n14

Struik, Dirk, 31112, 316, 336, 401n14

苏莱曼,47岁49岁

Süleyman, 47, 49

萨顿,托马斯,248

Sutton, Thomas, 248

Swirles,Bertha321,402 n3

Swirles, Bertha, 321, 402n3

瑞士联邦理工学院ETH),218,275,281,334

Swiss Federal Institute of Technology (ETH), 218, 275, 281, 334

西尔维斯特·詹姆斯,8 , 83 , 85 , 160

Sylvester, James, 8, 83, 85, 160

对称性:代数和,16;弯曲空间和,236;爱因斯坦和,306 14,322,399 n5 ;电磁学和,175 – 76;哈里奥特,353 n8 ;海维赛德和,376 n13 不变性和,90,192,236,268,305 诺特定理和,3067,311,33536,400 n7 模式,16 旋转反射 90,192,193 ;空间和90时空192 – 93,198,204,382 n27 和,248,268-71,301-2,308,314,393n20时间轴开启 335​

symmetry: algebra and, 16; curved space and, 236; Einstein and, 30614, 322, 399n5; electromagnetism and, 17576; Harriot and, 353n8; Heaviside and, 376n13; invariance and, 90, 192, 236, 268, 305; Noether theorems and, 3067, 311, 33536, 400n7; patterns of, 16; rotations and reflections and, 90, 192, 193; space and, 90; space-time and, 19293, 198, 204, 382n27; tensors and, 248, 26871, 3012, 308, 314, 393n20; timeline on, 335

泰特,玛格丽特·波特,145

Tait, Margaret Porter, 145

Tait , Peter Guthrie , 114 代数和,346 n3 ;背景11920;Campbell119,150,167Cayley137,156,160,163,179,187,189,191,194,206,333,378 n3 弯曲空间385 n12125,127,143,145,150,185,195 ​电磁和,11824,12829,137,139,142 – 45 ;元数初等论述147 – 50,332 ;高尔夫和,160,165;不变性19194 麦克斯韦118 24,12829,137,139,14245,152,167,331,366 n2,367 n7,369 n18,369 n21 ​nabla 和,14344146148150156161243;四元数和,114117146671701778118487;高级牧马人,119121热力学概论149372 n3 ;时空和,18995206;教学方法,145;张量和,243258; Thomson 和,142147501581778118619433234367 n7,371 n1,385 n12;时间线,33134自然哲学论文14849332;矢量场和,124139142146150155157

Tait, Peter Guthrie, 114: algebra and, 346n3; background of, 11920; Campbell and, 119, 150, 167; Cayley and, 137, 156, 160, 163, 179, 187, 189, 191, 194, 206, 333, 378n3; curved space and, 385n12; electricity and, 125, 127, 143, 145, 150, 185, 195; electromagnetism and, 11824, 12829, 137, 139, 14245; Elementary Treatise on Quaternions, 14750, 332; golf and, 160, 165; invariance and, 19194; Maxwell and, 11824, 12829, 137, 139, 14245, 152, 167, 331, 366n2, 367n7, 369n18, 369n21; nabla and, 14344, 146, 148, 150, 156, 161, 243; quaternions and, 114, 117, 14667, 170, 17781, 18487; as Senior Wrangler, 119, 121; Sketch of Thermodynamics, 149, 372n3; space-time and, 18995, 206; teaching methods of, 145; tensors and, 243, 258; Thomson and, 142, 14750, 158, 17781, 186, 194, 33234, 367n7, 371n1, 385n12; timeline on, 33134; A Treatise on Natural Philosophy, 14849, 332; vector field and, 124, 139, 142, 146, 150, 155, 157

帖木儿,49岁

Tamerlane, 49

切线36,222,225-26,315-16,383n6​​

tangents, 36, 222, 22526, 31516, 383n6

Tartaglia, Niccolò: 代数和, 12 , 347 n16; Cardano 和, 12 , 347 n16; 向量的思想和, 4649 , 352 n3

Tartaglia, Niccolò: algebra and, 12, 347n16; Cardano and, 12, 347n16; ideas for vectors and, 4649, 352n3

泰勒级数,54

Taylor series, 54

电报,119;电磁学和,95;Heaviside和169-72、177 工业革命和,83;国际,85;马可尼和,168;Øersted 和,9495128330;汤姆森和,1474816869177332;时间表,330332;惠斯通和,169

telegraphy, 119; electromagnetism and, 95; Heaviside and, 16972, 177; Industrial Revolution and, 83; international, 85; Marconi and, 168; Øersted and, 9495, 128, 330; Thomson and, 14748, 16869, 177, 332; timeline for, 330, 332; Wheatstone and, 169

积分:代数和,17弯曲空间和,219;发展,26,41 爱因斯坦和,28590,305,316,318;格罗斯曼28590索引符号26265里奇和,241,244,27174,303,305,316,318,333 – 34 时间轴333 34

tensor calculus: algebra and, 17; curved space and, 219; development of, 26, 41; Einstein and, 28590, 305, 316, 318; Grossmann and, 28590; index notation and, 26265; Ricci and, 241, 244, 27174, 303, 305, 316, 318, 33334; timeline on, 33334

TensorFlow,248

TensorFlow, 248

TensorLab,248

TensorLab, 248

Tensorly,248

Tensorly, 248

张量网络(TN),324

tensor networks (TN), 324

量:抽象概念和,254,265;代数和,242 43,246,258 65,284 86,289,295,300 算法 263 算术 255 ;人工智能(AI )249,387 n12 ;天文学276,293安奇恒等式31011,33536,372 n9,395 n35,400 n8,400 n10,401 n13 笛卡尔坐标和262,27072,284,290,295,391 n7,392 n14 ;交换245,268,286复数和,244,25152;的分量 214 15,226,245 – 48,25965,26771,28890,302,3078 计算能力26568 概念,ix – x 坐标变换242,25556,25965,26869,272,283,287,291,295,389 n22,390 n23,392 n14,394 n30 晶体和,322 卷曲和,243,271,287曲面242,244数据科学和,24755衍生264, 26972 , 283 , 286 , 28889 , 295 , 29830 1, 30912 , 386 n9, 389 n22, 390 n23, 397 n40; 微分学和, 24244 , 25658 , 26874 , 284 , 289 , 389 n22, 393 n19; 微分几何和, 272 , 289 , 393 n19; 狄拉克和, 251 , 253 , 323 ; 发散和, 243 , 264 , 29830 1;爱因斯坦和, xii24144,253,258,261,26671,27530 3,321,390 n2,391 nn6–10,392 nn11–14,393 n21,394 nn30–31,395 nn32–35,396 n37,397 n38,397 nn41–44,399 n46 电磁241,244,271,275,280,288 90,296 – 99,391 n8,397 n44;​​​​​​​​电子和,251,280;以太和,280,292欧几里得空间253,264 65,268 – 72,280,284,299,303;法拉第 289 ;通量279,289 90 四维数学247 – 48,286 – 88, 289 , 298 ;伽利略定律和, 277 , 390 n2, 393 n21;高斯和, 243 , 269 , 272 , 279 , 284 , 303 ;广义相对论和, 258 , 267 , 27430 3, 29830 3, 31419 , 323 , 389 n20, 391 n9, 392 n12, 392 n14, 394 n39, 395 nn32–35, 397 n44;几何和24142,245,259,272,28386,289,295 – 98,303 ;吉布斯 245 ;格拉德,243,264 格拉斯曼245,253,256,267;重力275 30 3,390 n2,390 n5,391 nn7–8,393 n21,397 n41 ;​​​Grossmann和244,253,258,275,282 95,298 – 99,303 Hamilton,243 45,267,298,395 n34 ;Harriot和,256,277Heaviside和,289 Hilbert和29498,301 重要性 322 – 24 指数符号258 – 64,264 65,268 – 71 积分,256,392 n16不变性24244,25559,26269,27273,285,290,389 n22,390 n23 ; 发明240 73;平方反比定律277,279,393 n21 ;克莱242,25556,27475,29395,298;拉格朗日和,279;拉普拉斯和391 n7 运动定律和271 列维-奇维塔定律274,285,289,294 98,303 ​洛伦兹和24243,262,269,27980,28384,288,292,300,392 n18;磁场矢量392 n18 矩阵 245 48,253 – 55,258 – 65,270,324,388 n19,389 n22 ​​​​390 n23, 392 n18; Maxwell 和, 238 , 24143 , 256 , 264 , 271 , 273 , 27980 , 28790 , 300 , 387 n10; 美索不达米亚和, 255 ; Minkowski 和, 24243 , 253 , 255 , 26869 , 28089 , 293 , 29899 , 303 , 392 n18, 397 nn39–40, 400 n9; 动量和, 29091 , 295 , 29830 0, 321 ;运动与25758,264,271 – 76,283,295 – 96,299;命名,244 – 46 牛顿271,275 80,283,285 87,291,295,3013,391 nn7–8,395 n32,397 n41,397 n44 NLP,249 – 51,387 n12​符号和,23339251542586627328889298390 n23,400 ;平行四边形规则和,24525531419;行星和,27576;庞加莱和,24228081 ; x的幂;毕达哥拉斯和,243;二次方程和243 257;量子理论和,24755265292302388 n14,389 n20;四元数和,243 – 44 258267无线电波和,241;实在和,271;里奇和,2404625375285872902949529830 3,320 ,323325 333393 n21,397 n42,400 n10 ;黎曼和,23339,243 – 44,248,255,257,260 – 61,267,269,273,283,28687,295,303,310 11,332,386 n9 机器人和 259 ;旋转和,259 – 69,389 nn21–22说得更多212 – 16 标量数和,243246252552592627027980286301324389 n20,389 n22,397 n39,397 n43;信号处理和,247;作为更简单的方法,30912 ;索末菲和,212132879229529899;狭义相对论和,2752788529192299;对称性和,248268713012393 n20 ;Tait 和,243258;Thomson 和,273;三维空间和,254267287;拓扑和,11122633;矢量场和,264280287;速度和,x41283287392 n18 ;迫切需要,32223

tensors: abstract concepts and, 254, 265; algebra and, 24243, 246, 25865, 28486, 289, 295, 300; algorithms and, 263; arithmetic and, 255; artificial intelligence (AI) and, 249, 387n12; astronomy and, 276, 293; Bianchi identities and, 31011, 33536, 372n9, 395n35, 400n8, 400n10, 401n13; Cartesian coordinates and, 262, 27072, 284, 290, 295, 391n7, 392n14; commutative law and, 245, 268, 286; complex numbers and, 244, 25152; components of, 21415, 226, 24548, 25965, 26771, 28890, 302, 3078; computational power of, 26568; concept of, ixx; coordinate transformations and, 242, 25556, 25965, 26869, 272, 283, 287, 291, 295, 389n22, 390n23, 392n14, 394n30; crystallography and, 322; curl and, 243, 271, 287; curved surfaces and, 242, 244; data science and, 24755; derivatives and, 264, 26972, 283, 286, 28889, 295, 298301, 30912, 386n9, 389n22, 390n23, 397n40; differential calculus and, 24244, 25658, 26874, 284, 289, 389n22, 393n19; differential geometry and, 272, 289, 393n19; Dirac and, 251, 253, 323; divergence and, 243, 264, 298301; Einstein and, xii, 24144, 253, 258, 261, 26671, 275303, 321, 390n2, 391nn6–10, 392nn11–14, 393n21, 394nn30–31, 395nn32–35, 396n37, 397n38, 397nn41–44, 399n46; electromagnetism and, 241, 244, 271, 275, 280, 28890, 29699, 391n8, 397n44; electrons and, 251, 280; ether and, 280, 292; Euclidean space and, 253, 26465, 26872, 280, 284, 299, 303; Faraday tensor, 289; flux and, 279, 28990; four-dimensional mathematics and, 24748, 28688, 289, 298; Galileo’s law and, 277, 390n2, 393n21; Gauss and, 243, 269, 272, 279, 284, 303; general theory of relativity and, 258, 267, 274303, 298303, 31419, 323, 389n20, 391n9, 392n12, 392n14, 394n39, 395nn32–35, 397n44; geometry and, 24142, 245, 259, 272, 28386, 289, 29598, 303; Gibbs and, 245; grad and, 243, 264; Grassmann and, 245, 253, 256, 267; gravity and, 275303, 390n2, 390n5, 391nn7–8, 393n21, 397n41; Grossmann and, 244, 253, 258, 275, 28295, 29899, 303; Hamilton and, 24345, 267, 298, 395n34; Harriot and, 256, 277; Heaviside and, 289; Hilbert and, 29498, 301; importance of, 32224; index notation and, 25864, 26465, 26871; integrals and, 256, 392n16; invariance and, 24244, 25559, 26269, 27273, 285, 290, 389n22, 390n23; inventing, 24073; inverse square law and, 277, 279, 393n21; Klein and, 242, 25556, 27475, 29395, 298; Lagrange and, 279; Laplace and, 391n7; laws of motion and, 271; Levi-Civita and, 274, 285, 289, 29498, 303; Lorentz and, 24243, 262, 269, 27980, 28384, 288, 292, 300, 392n18; magnetic field vector and, 392n18; matrices and, 24548, 25355, 25865, 270, 324, 388n19, 389n22, 390n23, 392n18; Maxwell and, 238, 24143, 256, 264, 271, 273, 27980, 28790, 300, 387n10; Mesopotamia and, 255; Minkowski and, 24243, 253, 255, 26869, 28089, 293, 29899, 303, 392n18, 397nn39–40, 400n9; momentum and, 29091, 295, 298300, 321; motion and, 25758, 264, 27176, 283, 29596, 299; naming of, 24446; Newton and, 271, 27580, 283, 28587, 291, 295, 3013, 391nn7–8, 395n32, 397n41, 397n44; NLP and, 24951, 387n12; notation and, 23339, 25154, 25866, 273, 28889, 298, 390n23, 400; parallelogram rule and, 245, 255, 31419; planets and, 27576; Poincaré and, 242, 28081; power of, x; Pythagoras and, 243; quadratic equations and, 243, 257; quantum theory and, 24755, 265, 292, 302, 388n14, 389n20; quaternions and, 24344, 258, 267; radio waves and, 241; reality and, 271; Ricci and, 24046, 25375, 28587, 290, 29495, 298303, 320, 323, 325, 333, 393n21, 397n42, 400n10; Riemann and, 23339, 24344, 248, 255, 257, 26061, 267, 269, 273, 283, 28687, 295, 303, 31011, 332, 386n9; robots and, 259; rotation and, 25969, 389nn21–22; saying more with, 21216; scalar numbers and, 243, 246, 25255, 259, 26270, 27980, 286, 301, 324, 389n20, 389n22, 397n39, 397n43; signal processing and, 247; as simpler method, 30912; Sommerfeld and, 21213, 28792, 295, 29899; special theory of relativity and, 275, 27885, 29192, 299; symmetry and, 248, 26871, 3012, 393n20; Tait and, 243, 258; Thomson and, 273; three-dimensional space and, 254, 267, 287; topology and, 111, 22633; vector field and, 264, 280, 287; velocity and, x, 41, 283, 287, 392n18; vital need for, 32223

萨克雷,威廉·梅克皮斯,149

Thackeray, William Makepeace, 149

热力学, 14849,17778,371 n3

thermodynamics, 14849, 17778, 371n3

托马斯·哈里奥特:科学人生(Arianrhod),347 n10

Thomas Harriot: A Life in Science (Arianrhod), 347n10

汤姆森,威廉(开尔文勋爵):弯曲空间和,236;不变性和,194;开尔文温标,148;麦克斯韦和,12224129131136140142367 n7;北大西洋电报公司和,148;四元数和,14751155158591687217781186;时空和,19421315; Tait142,14750,158,17781,186,194,33234,367 n7,371 n1,385 n12 ;电报147 – 48,168 – 69,177,332 ;和,273 ;时间轴,33133 自然哲学论文14849,332

Thomson, William (Lord Kelvin): curved space and, 236; invariance and, 194; Kelvin temperature scale of, 148; Maxwell and, 12224, 129, 131, 136, 140, 142, 367n7; North Atlantic Telegraph Company and, 148; quaternions and, 14751, 155, 15859, 16872, 17781, 186; space-time and, 194, 21315; Tait and, 142, 14750, 158, 17781, 186, 194, 33234, 367n7, 371n1, 385n12; telegraphy and, 14748, 16869, 177, 332; tensors and, 273; timeline on, 33133; A Treatise on Natural Philosophy, 14849, 332

亨利·戴维·梭罗,84岁331

Thoreau, Henry David, 84, 331

三维空间代数14; 弯曲空间21923,229 汉密尔顿和,14,6869,7476,9091,9799,1067,114,16162,183向量的想法68MRI97数和104,107,114,161 – 62,178,183 ;​旋转,2,68,75,90-94,97,104,107,114,162,183,345 n1 时空191-92,206-12,207,215和,254,267,287

three-dimensional space: algebra and, 14; curved space and, 21923, 229; Hamilton and, 14, 6869, 7476, 9091, 9799, 1067, 114, 16162, 183; ideas for vectors and, 68; MRI and, 97; quaternions and, 104, 107, 114, 16162, 178, 183; rotation in, 2, 68, 75, 9094, 97, 104, 107, 114, 162, 183, 345n1; space-time and, 19192, 20612, 207, 215; tensors and, 254, 267, 287

底格里斯河,xiv

Tigris River, xiv

时间机器(威尔斯),207

Time Machine, The (Wells), 207

拓扑学,111;曲面和,22633 ;不变性和,22630;彭罗斯和228,237

topology, 111; curved surfaces and, 22633; invariance and, 22630; Penrose and, 228, 237

轨迹,35,46-49,181-82,274,277

trajectories, 35, 4649, 18182, 274, 277

天体力学论文(拉普拉斯)65,104-5,126,329

Treatise on Celestial Mechanics (Laplace), 65, 1045, 126, 329

电磁论(麦克斯韦),145;微积分与,369 n19,369 n21,349 n2 ;微分与,369 n21;影响,343 n2;积分与,369 n19;平方反比定律与,367 n11;四元数与,150 153 154 156 157 159 61 168 170 ,174 176 79 ,183 ,365 n14,372 n7,373 n12 时空与,213 汤姆森与,367 n7;时间轴332

Treatise on Electricity and Magnetism (Maxwell), 145; calculus and, 369n19, 369n21, 349n2; differentials and, 369n21; impact of, 343n2; integrals and, 369n19; inverse square law and, 367n11; quaternions and, 150, 153, 154, 156, 157, 15961, 168, 170, 174, 17679, 183, 365n14, 372n7, 373n12; space-time and, 213; Thomson and, 367n7; timeline on, 332

自然哲学论文集》(Tait and Thomson),14849,332

Treatise on Natural Philosophy, A (Tait and Thomson), 14849, 332

三角学,xvxvi90,173,328,344 n4,344 n7​​​

trigonometry, xvxvi, 90, 173, 328, 344n4, 344n7

剑桥大学三一学院, 72 , 81 , 120 , 124 , 167 , 231

Trinity College, Cambridge, 72, 81, 120, 124, 167, 231

都柏林圣三一学院,6566,346 n4

Trinity College, Dublin, 6566, 346n4

三等学位,119,121,160

Tripos, 119, 121, 160

Ṭūsī Sharaf al-Dīn al-, 11 , 13 , 328

Ṭūsī Sharaf al-Dīn al-, 11, 13, 328

实验18,97,141,330

two-slit experiment, 18, 97, 141, 330

廷德尔,约翰331,359 n19

Tyndall, John, 331, 359n19

Uhlenbeck,George ,96,335,362 n27

Uhlenbeck, George, 96, 335, 362n27

乌克兰,317

Ukraine, 317

英国xix117,333

United Kingdom, xix, 117, 333

伦敦大学学院,72,160

University College, London, 72, 160

大学女子学院xix188-89,321

University Women’s Colleges, xix, 18889, 321

Unruh ,William 264,389 n20

Unruh, William, 264, 389n20

天王星,62

Uranus, 62

大熊星座,xvii

Ursa Major, xvii

四元数在物理学中的应用(McAulay),183

Utility of Quaternions in Physics (McAulay), 183

矢量委员会,21112

Vector Commission, 21112

矢量:弯曲空间和,237电磁3,118-45法拉第133-40 麦克斯韦3,110,124,139-42,150-55,157,159,166,183,201-4,215,237,264,287-89,325,331-32,343n2四元数和15255、159、175;时空和2012;Tait124、139、142、146、150、155、157 264、280、287

vector field: curved space and, 237; electromagnetism and, 3, 11845; Faraday and, 13340; Maxwell and, 3, 110, 124, 13942, 15055, 157, 159, 166, 183, 2014, 215, 237, 264, 28789, 325, 33132, 343n2; quaternions and, 15255, 159, 175; space-time and, 2012; Tait and, 124, 139, 142, 146, 150, 155, 157; tensors and, 264, 280, 287

向量:代数和,1 17 参见代数积分 18 41 参见积分;术语的创造 xi xii 85、246、24954、263 – 66、270 分量和,xxixxv3536、4445、4748、52、7779、179、18919 0、193、201 – 2 概念ixx; 交叉积和,7880,104,106,144,155,162,17980,213,245 弯曲空间和,217 – 39;作为数据存储,xxv 77 100 爱因斯坦和207,211;四维的, 20512吉布斯和,78,177 78 汉密尔顿命名76 海维赛德和,78,143,17177 想法42 – 68 ;重要性,322 24;磁场,1552012264287392 n18;大小,60208;麦克斯韦和,11845;符号和,xxxxvi;位置,xx449092208221259391 n7;幂,x15760;四元数和,1001714687;乘法的右手法则,279155;空间和,69100;时空和, 18821 6 ;张量和,24030 3 (另请参阅张量);用途,xxivxxvi;迫切需要,32223;“战争” ,17986;整体向量与分量,78139144148159183

vectors: algebra and, 117 (see also algebra); calculus and, 1841 (see also calculus); coining of term, xixii; column, 85, 246, 24954, 26366, 270; components and, xxixxv, 3536, 4445, 4748, 52, 7779, 179, 189190, 193, 2012; concept of, ixx; cross products and, 7880, 104, 106, 144, 155, 162, 17980, 213, 245; curved space and, 21739; as data storage, xxv, 77100; Einstein and, 207, 211; four-dimensional, 20512; Gibbs and, 78, 17778; Hamilton’s naming of, 76; Heaviside and, 78, 143, 17177; ideas for, 4268; importance of, 32224; magnetic field, 155, 2012, 264, 287, 392n18; magnitude of, 60, 208; Maxwell and, 11845; notation and, xxxxvi; position, xx, 44, 90, 92, 208, 221, 259, 391n7; power of, x, 15760; quaternions and, 10017, 14687; right-hand rule for multiplying, 2, 79, 155; space and, 69100; space-time and, 188216; tensors and, 240303 (see also tensors); uses of, xxivxxvi; vital need for, 32223; “wars” over, 17986; whole vectors versus components, 78, 139, 144, 148, 159, 183

向量空间:四元和,107,163,362 n29 ;跨度,77 张量和,247,252,265时间轴331,333

vector space: quaternions and, 107, 163, 362n29; spanning, 77; tensors and, 247, 252, 265; timeline on, 331, 333

速度:微积分和,32,41,352 n15;弯曲空间和,230爱因斯坦和,3056;四,xxiv345 n8;向量概念和,43,50,52 ;麦克斯韦和,130 31,136 符号和,xxi;四元和,146,156,172,182 时空和379 n10 和,x 41,283,287,392 n18

velocity: calculus and, 32, 41, 352n15; curved space and, 230; Einstein and, 3056; four, xxiv, 345n8; ideas for vectors and, 43, 50, 52; Maxwell and, 13031, 136; notation and, xxi; quaternions and, 146, 156, 172, 182; space-time and, 379n10; tensors and, x, 41, 283, 287, 392n18

委罗内塞,朱塞佩,25758

Veronese, Guiseppe, 25758

Viète,François,1516 岁

Viète, François, 1516

“牧马人的愿景,A”(麦克斯韦),12021

“Vision of a Wrangler, A” (Maxwell), 12021

沃尔德马·福伊特,205 , 24445

Voigt, Woldemar, 205, 24445

亚历山德罗·沃尔特,12425 , 330

Volta, Alessandro, 12425, 330

电压,125,152,157,172

voltage, 125, 152, 157, 172

伏尔泰( 3738)

Voltaire, 3738

瓦尔登84,331

Walden Pond, 84, 331

Wallis , John代数和,1415348 n18;无限算术3234;背景,3233;微积分和,27,3134,350 nn7–8 Harriot1415,27,31 – 32,52,61,111,348 n18,350 n7,354 n10 向量思想和52 – 53,58,61,354 n11 ,111

Wallis, John: algebra and, 1415, 348n18; Arithmetica Infinitorum, 3234; background of, 3233; calculus and, 27, 3134, 350nn7–8; Harriot and, 1415, 27, 3132, 52, 61, 111, 348n18, 350n7, 354n10; ideas for vectors and, 5253, 58, 61, 354n11; quaternions and, 111

沃伦,约翰,58 , 60 , 104 , 113 , 330

Warren, John, 58, 60, 104, 113, 330

Wathaurong 国家,xvii

Wathaurong country, xvii

瓦特,詹姆斯,157

Watt, James, 157

韦伯,威廉,131

Weber, Wilhelm, 131

韦尔斯,HG,207

Wells, H. G., 207

韦塞尔,卡斯帕,58 , 60 , 330

Wessel, Caspar, 58, 60, 330

韦尔,赫尔曼,306

Weyl, Hermann, 306

查尔斯·惠斯通,169

Wheatstone, Charles, 169

威廉·惠威尔101,331

Whewell, William, 101, 331

惠特曼,安,38岁

Whitman, Ann, 38

安德鲁·威尔斯57 岁

Wiles, Andrew, 57

沃尔夫,伯特兰,365 n14

Wolff, Bertrand, 365n14

弗吉尼亚州伍尔夫,189

Woolf, Virginia, 189

应用4、7 – 11

word problems, 4, 711

华兹华斯,威廉,64 , 75 , 331

Wordsworth, William, 64, 75, 331

工人学院,xix123

Working Men’s College, xix, 123

第一次世界大战,95、293、317、334-35

World War I, 95, 293, 317, 33435

Wurdi Youang,xvii

Wurdi Youang, xvii

偏航,93

yaw, 93

Young, Thomas: 干涉图样和, 19 , 97 , 199 ; 光和, 1819 , 97 , 141 , 199 , 330 ; 双缝实验, 18 , 97 , 141 , 330

Young, Thomas: interference patterns and, 19, 97, 199; light and, 1819, 97, 141, 199, 330; two-slit experiment of, 18, 97, 141, 330

杨,威廉,242

Young, William, 242

Zangger,Heinrich,403 n5

Zangger, Heinrich, 403n5

芝诺,2631 , 40 , 44

Zeno, 2631, 40, 44